Вспомогательные функции

library(Rssa)
Предупреждение: пакет ‘Rssa’ был собран под R версии 4.2.3
Загрузка требуемого пакета: svd
Предупреждение: пакет ‘svd’ был собран под R версии 4.2.3
Загрузка требуемого пакета: forecast
Предупреждение: пакет ‘forecast’ был собран под R версии 4.2.3

Присоединяю пакет: ‘Rssa’

Следующий объект скрыт от ‘package:stats’:

    decompose
library(signal)
Предупреждение: пакет ‘signal’ был собран под R версии 4.2.3

Присоединяю пакет: ‘signal’

Следующий объект скрыт от ‘package:Rssa’:

    roots

Следующие объекты скрыты от ‘package:stats’:

    filter, poly
library(gsignal)
Предупреждение: пакет ‘gsignal’ был собран под R версии 4.2.3
Registered S3 methods overwritten by 'gsignal':
  method         from  
  plot.grpdelay  signal
  plot.specgram  signal
  print.freqz    signal
  print.grpdelay signal
  print.impz     signal
  print.specgram signal

Присоединяю пакет: ‘gsignal’

Следующий объект скрыт ‘.GlobalEnv’:

    dftmtx

Следующий объект скрыт от ‘package:signal’:

    Arma, as.Arma, as.Zpg, bartlett, bilinear, blackman, boxcar, butter, buttord, cheb1ord, chebwin, cheby1, cheby2,
    chirp, conv, decimate, ellip, ellipord, fftfilt, filter, filtfilt, fir1, fir2, flattopwin, freqs, freqs_plot,
    freqz, freqz_plot, gausswin, grpdelay, hamming, hanning, ifft, impz, interp, kaiser, kaiserord, levinson, Ma,
    medfilt1, poly, remez, resample, sftrans, sgolay, sgolayfilt, specgram, triang, unwrap, Zpg, zplane

Следующие объекты скрыты от ‘package:stats’:

    filter, gaussian, poly
source("eossa_new.R")


dftmtx <- function(n) {
  y <- stats::mvfft(diag(1, n))
  y
}

diag_averaging <- function(A){
  B <- A[nrow(A):1, ] |> Re()
  lapply(split(B, -(row(B) - col(B)) ), mean) |> as.numeric()
}

shift_vector <- function(vec) {
  last_element <- tail(vec, 1)
  vec <- vec[-length(vec)]
  shifted_vec <- c(last_element, vec)
  return(shifted_vec)
}

extend <- function(x, H){
  # Вычисление коэффициентов AR модели для дифференцированного ряда
  N <- length(x)
  p <- floor(N / 3)
  dx <- diff(x)
  # A <- ar(dx, order.max = p, method = "yule-walker")$ar
  A <- aryule(dx, p)$a
  
  # Правое расширение
  y <- x
  dy <- diff(y)
  er <- signal::filter(A, 1, dy)
  dy <- signal::filter(1, A, c(er, rep(0, H)))
  y <- y[1] + c(0, cumsum(dy))
  
  # Левое расширение
  y <- rev(y)
  dy <- diff(y)
  er <- signal::filter(A,1,dy)
  dy <- signal::filter(1,A,c(er, rep(0, H)))
  y <- y[1] + c(0, cumsum(dy))
  
  # Расширенный ряд
  xe <- rev(y)
  
  # Вывод результатов
  xe 
}

CiSSA

Подаётся на вход временной ряд, длина окна (если её нет, то она равна длине ряда + 1 пополам) и информация о том, нужно ли расширить ряд. Расширять ряд стоит при стохастическом тренде (Autoregressive extension (default). It is indicated for stationary and stochastic trend time series as well). Реализовано только Autoregressive extension.


На выходе список выдаётся список list(t_series, importance).
t_series — матрица, по столбцам которой располагаются временные ряды, отвечающие за частоты (i-1)/L, где i — номер столбца, L — длина окна.
importance — вектор, отвечающий за значимость i-ого временного ряда в разлолжении. Чем больше значение, тем больший вклад внёс i-тый временной ряд.

circulant_SSA <- function(ts, L = NULL, extend_flag = FALSE){
  time_series <- ts
  # Construct trajectory matrix
  N <- length(time_series)
  if (is.null(L)){
    L <- (N + 1)%/%2
  }
  # Проверка на расширения ряда
  if (extend_flag == FALSE){
    H <- 0
    time_series <- ts
  }
  else{
    H <- L
    time_series <- extend(ts, H)
  }
  
  X <- hankel(time_series, L)
  
  # Number of symmetric frequency pairs around 1/2
  if (L %% 2) {
    nf2 <- (L + 1) / 2 - 1
  } else {
    nf2 <- L / 2 - 1
  }
  
  # Number of frequencies <= 1/2
  nft <- nf2 + abs((L %% 2) - 2)
  
  # Decomposition
  # Estimate autocovariance     OK
  autocov <- numeric(L)
  for (m in 0:(L-1)){
    autocov[[m+1]] <- sum(time_series[1:(N-m)] * time_series[(1+m):N]) / (N-m)
  }
  
  # First row of circulant matrix
  circ_first_row <- numeric(L)
  for (m in 0:(L-1)){
    circ_first_row[[m+1]] <- (L-m)/L * autocov[[m+1]] + (m)/L * autocov[[L-m]]
  }
  
  # Build circulant matrix
  S_C <- matrix(circ_first_row, nrow = 1)
  shifted_vector <- circ_first_row
  for (i in 2:(L)) {
    shifted_vector <- shift_vector(shifted_vector)
    # S_C <- rbind(S_C, as.vector(shifted_vector))
    S_C <- rbind(as.vector(shifted_vector), S_C)
  }
  
  # Eigenvectors of circulant matrix (unitary base)
  U <- dftmtx(L)/sqrt(L)
  
  # Real eigenvectors (orthonormal base)
  U[, 1] <- Re(U[, 1])
  for (k in 1:nf2) {
    u_k <- U[, k + 1]
    U[, k + 1] <- sqrt(2) * Re(u_k)
    U[, L + 2 - (k + 1)] <- sqrt(2) * Im(u_k)
  }
  if (L %% 2 != 0) {
    U[, nft] <- Re(U[, nft])
  }
  
  # Eigenvalues of circulant matrix: estimated power spectral density
  psd <- abs(diag(t(U) %*% S_C %*% U))
  
  # Principal components
  W <- t(U) %*% X
  # Reconstruction
  # Elementary reconstructed series
  R <- matrix(0, nrow = N+2*H, ncol = L)
  for (k in 1:L) {
    R[, k] <- U[ ,k] %*% t(W[k, ]) |> diag_averaging()
  }
  
  # Grouping by frequency
  # Elementary reconstructed series by frequency
  Z <- matrix(0, nrow = N+2*H, ncol = nft)
  Z[, 1] <- R[, 1]
  # Importance of component
  imp <- numeric(nft)
  lambda_sm <- sum(psd)
  imp[1] <- psd[1]/lambda_sm
  for (k in 1:nf2) {
    Z[, k + 1] <- R[, k + 1] + R[, L + 2 - (k + 1)]
    imp[k+1] <- (psd[k+1] + psd[ L + 2 - (k + 1)])/lambda_sm
  }
  if (L %% 2 != 0) {
    Z[, nft] <- R[, nft]
    imp[nft] <- psd[nft] / lambda_sm
  }
  
  list(t_series = Z[(H+1):(N+H),],
       importance = imp,
       freq = (0:(length(imp) -1))/L
       )
}
# groups - list of frequencies
grouping_cissa <- function(cissa_res, groups){
  freq <- cissa_res$freq
  t_series <- cissa_res$t_series
  
  residuals <- 0
  result <- setNames(as.list(rep(0, length(groups))), names(groups))
  for (i in 1:length(cissa_res$freq)){
    flag <- FALSE
    for (name in names(groups)){
      if (groups[[name]][1] <= freq[i] & freq[i] <= groups[[name]][2]){
        flag <- TRUE
        result[[name]] <- result[[name]] + t_series[, i]
      }
    }
    
    if (flag == FALSE){
      residuals <- residuals + t_series[, i]
    }
  }
  
  result[["residuals"]] <- residuals
  result
}
generate_ts <- function(func, n=1e3, ...){
  1:n |> func(...) |> ts()
}

f_cos <- function(x, A = 1, omega = 1/4, phi = 0){
  f_exp_mod_harm_series(x, A, alpha = 0, omega = omega, phi = phi)
}

f_sin <- function(x, A = 1, omega = 1/4, phi = 3*pi/2){
  f_exp_mod_harm_series(x, A, alpha = 0, omega = omega, phi = phi)
}

f_exp <- function(x, A = 1, alpha = 1){
  A * exp(alpha * x)
}

f_exp_cos <- function(x, A = 1, alpha = 1, omega = 1/4, phi = 0){
  f_exp_mod_harm_series(x, A, alpha, omega, phi)
}

f_const <- function(x, C = 0){
  rep(C, length(x))
}

f_exp_mod_harm_series <- function(x, A = 1, alpha = 1, omega = 1/4, phi = 0){
  A*exp(alpha*x)*cos(2*pi*omega*x + phi)
}

f_linear <- function(x, a = 1, b = 0){
  a*x + b
}
mse <- function(f_true, f_reconstructed){
   mean((f_true - f_reconstructed)^2) 
}

Ошибка при Lw in N, Kw not in N

n <- 96*2+5
L <- 96
f_sum <- function(x){
  f_const(x, C = 1) + f_cos(x, omega = 1/12) 
}


f_const |> generate_ts(n, C = 1) |>
  plot(col = "green", ylim = c(-1, 2), ylab = "f_n")
f_cos |>
  generate_ts(n, omega = 1/12) |>
  lines(col="green")
f_sum |> generate_ts(n) |> lines(lwd = 3, col='red')
f_n <- f_sum(1:n)



c <- circulant_SSA(f_n, L = 96, extend_flag = FALSE)
r <- grouping_cissa(c,
               groups = list(
                 trend = c(0, 1/100),
                 sesonal = c(1/99, 1/10)
               )
               )

f_C <- f_const |> generate_ts(n, C = 1)
f_c <- f_cos |> generate_ts(n, omega = 1/12)
print("Ошибки при CiSSA")
[1] "Ошибки при CiSSA"
print(paste("Ошибка при вычислении C = 1: ", mse(f_C, r$trend) |> format(scientific = TRUE, digits = 2) ))
[1] "Ошибка при вычислении C = 1:  3.2e-31"
print(paste("Ошибка при вычислении cos(pi/12): ", mse(f_c, r$sesonal) |> format(scientific = TRUE, digits = 2) ))
[1] "Ошибка при вычислении cos(pi/12):  3.6e-30"
lines(1:n, r$trend, col="blue")
lines(1:n, r$sesonal, col="blue")


f_const |> generate_ts(n, C = 1) |>
  plot(col = "green", ylim = c(-1, 2), ylab = "f_n")
f_cos |>
  generate_ts(n, omega = 1/12) |>
  lines(col="green")
f_sum |> generate_ts(n) |> lines(lwd = 3, col='red')
f_n <- f_sum(1:n)

s <- ssa(f_n, L = 96)
r <- reconstruct(s, groups=list(
  trend = 1,
  sesonal = 2:3
))


print("Ошибки при SSA")
[1] "Ошибки при SSA"
print(paste("Ошибка при вычислении C = 1: ", mse(f_C, r$trend) |> format(scientific = TRUE, digits = 2)  ))
[1] "Ошибка при вычислении C = 1:  9.6e-05"
print(paste("Ошибка при вычислении cos(pi/12): ", mse(f_c, r$sesonal) |> format(scientific = TRUE, digits = 2)))
[1] "Ошибка при вычислении cos(pi/12):  9.6e-05"
lines(1:n, r$trend)
lines(1:n, r$sesonal)

Проверка разделимости непериодических компонент + автогруппировка SSA

n <- 96*2-1
L <- 96

C <- 1
omega_cs <- 1/12
omega_sn <- 1/24
a <- 1/100
f_sum <- function(x){
  f_const(x, C = C) +
    f_cos(x, omega = omega_cs) +
    f_exp(x, a = a) +
    f_sin(x, omega = omega_sn)
}


f_C <- f_const |> generate_ts(n, C = C)
f_c <- f_cos |> generate_ts(n, omega = omega_cs)
f_s <- f_sin |> generate_ts(n, omega = omega_sn)
f_e <- f_exp |> generate_ts(n, a = a)

f_n <- f_sum(1:n)

library(xtable)
Предупреждение: пакет ‘xtable’ был собран под R версии 4.2.3
# Шаг 2: Создание примера данных
data <- data.frame(
  Метод = c("SSA", "CiSSA"),
  e_err = c(20, 20),
  c_err = c(23, 35),
  ec_err = c(20, 20),
  sin_err = c (20, 20),
  cos_err = c(1, 1)
)


# Отрисовка ряда f_n
plot(f_n, type = "l", lwd = 3, col = 'red', ylim = c(-2, 10),
     xlab = "Время", ylab = "Значения ряда", main = "Разложение временного ряда")

# Добавление отдельных компонентов (f_C, f_c, f_e)
lines(f_C, col = "blue")  # Компонент f_C
lines(f_c, col = "blue")  # Компонент f_c
lines(f_e, col = "blue")  # Компонент f_e
lines(f_s, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)









c <- circulant_SSA(f_n, L = L, extend_flag = TRUE)
# r <- c$t_series
r <- grouping_cissa(c,
                    groups = list(
                      # trend = c(0, 1/100),
                      trend = c(0, 1/1000),
                      sesonal_cos = c(1/14, 1/10),
                      sesonal_sin = c(1/26, 1/23)
                    ))

data$cos_err[2] <- mse(f_s, r$sesonal_sin) |> formatC(format = "e", digits = 1)
data$sin_err[2] <- mse(f_c, r$sesonal_cos) |> formatC(format = "e", digits = 1)
data$ec_err[2] <- mse(f_C+f_e, r$trend) |> formatC(format = "e", digits = 1)


# png("C:/Users/nik1m/Desktop/уник/6 сем/курсач/Текст работы/img/trend inseparability/CiSSA.png")  # сохранение в формате PNG

plot(1:n, f_n, type = "l", lwd=3, ylim= c(-2, 10), col="red",
     xlab = "Время", ylab = "Значения ряда", main = "CiSSA разложение временного ряда")
lines(1:n, r$trend, col = "blue")
lines(1:n, r$sesonal_sin, col = "blue")
lines(1:n, r$sesonal_cos, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)


# dev.off()  # завершение сохранения










s <- ssa(f_n, L)
e <- eossa(s, 1:10, k = 7)

g_sesonal <- grouping.auto(e, base = "eigen",
                   freq.bins = list(trend = c(0.001),
                                    sesonal2 = c(1/25, 1/23),
                                    sesonal1 = c(1/13, 1/11)
                                    ),
                   threshold = 0.1)


r <- Rssa::reconstruct(e, groups=c(list(exp = 1,
                                C = 2
                                ),
                             g_sesonal)
                 )

plot(wcor(e, groups = 1:24), scales = list(at = c(10, 20, 30)))
Предупреждение в wcor.ossa(e, groups = 1:24) :
  Component matrices are not F-orthogonal (max F-cor is 0.93). W-cor matrix can be irrelevant

data$c_err[1] <- mse(f_C, r$C) |> formatC(format = "e", digits = 1)
data$e_err[1] <- mse(f_e, r$exp) |> formatC(format = "e", digits = 1)
data$cos_err[1] <- mse(f_c, r$sesonal1) |> formatC(format = "e", digits = 1)
data$sin_err[1] <- mse(f_s, r$sesonal2) |> formatC(format = "e", digits = 1)
data$ec_err[1] <- mse(f_C+f_e, r$C+r$exp) |> formatC(format = "e", digits = 1)


# png("C:/Users/nik1m/Desktop/уник/6 сем/курсач/Текст работы/img/trend inseparability/SSA.png")  # сохранение в формате PNG

plot(1:n, f_n, type = "l", lwd=3, ylim= c(-2, 10), col="red",
     xlab = "Время", ylab = "Значения ряда", main = "SSA разложение временного ряда")

lines(1:n, r$trend, type = "l", col="green")
lines(1:n, r$exp, type = "l", ylim= c(-2, 10), col="blue")
lines(1:n, r$C, col = "blue")
lines(1:n, r$sesonal1, col = "blue")
lines(1:n, r$sesonal2, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)








# Шаг 3: Преобразование данных в формат LaTeX
table_latex <- xtable(data, caption = "Example Table")

# Шаг 4: Вывод таблицы в LaTeX файл
print(table_latex, include.rownames = FALSE)
% latex table generated in R 4.2.2 by xtable 1.8-4 package
% Mon Nov  4 13:25:09 2024
\begin{table}[ht]
\centering
\begin{tabular}{llllll}
  \hline
Метод & e\_err & c\_err & ec\_err & sin\_err & cos\_err \\ 
  \hline
SSA & 2.2e-25 & 2.2e-25 & 4.2e-28 & 3.8e-29 & 1.6e-29 \\ 
  CiSSA & 20 & 35 & 3.5e-02 & 1.4e-04 & 1.9e-03 \\ 
   \hline
\end{tabular}
\caption{Example Table} 
\end{table}

Пример cos*exp

n <- 96*2-1
L <- 96
eps <- 1/(L+1)

C <- 1
omega_cs <- 1/12
omega_sn <- 1/24
a <- 1/100
omega_exp <- 1/48
f_sum <- function(x){
    f_cos(x, omega = omega_cs) +
    f_exp_mod_harm_series(x, a = a, omega = omega_exp) +
    f_sin(x, omega = omega_sn)
}


f_c <- f_cos |> generate_ts(n, omega = omega_cs)
f_s <- f_sin |> generate_ts(n, omega = omega_sn)
f_e <- f_exp_mod_harm_series |> generate_ts(n, a = a, omega = omega_exp)

f_n <- f_sum(1:n)

library(xtable)

# Шаг 2: Создание примера данных
data <- data.frame(
  Метод = c("SSA", "CiSSA"),
  exp_err = c(20, 20),
  sin_err = c (20, 20),
  cos_err = c(1, 1)
)


# Отрисовка ряда f_n
plot(f_n, type = "l", lwd = 3, col = 'red', ylim = c(-10, 10),
     xlab = "Время", ylab = "Значения ряда", main = "Разложение временного ряда")

# Добавление отдельных компонентов (f_C, f_c, f_e)
lines(f_c, col = "blue")  # Компонент f_c
lines(f_e, col = "blue")  # Компонент f_e
lines(f_s, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)









c <- circulant_SSA(f_n, L = L, extend_flag = TRUE)
# r <- c$t_series
r <- grouping_cissa(c,
                    groups = list(
                      trend = c(0, 1/26-eps),
                      sesonal_cos = c(1/14, 1/10),
                      sesonal_sin = c(1/26, 1/23)
                    ))

data$cos_err[2] <- mse(f_s, r$sesonal_sin) |> formatC(format = "e", digits = 1)
data$sin_err[2] <- mse(f_c, r$sesonal_cos) |> formatC(format = "e", digits = 1)
data$exp_err[2] <- mse(f_e, r$trend) |> formatC(format = "e", digits = 1)


# png("C:/Users/nik1m/Desktop/уник/6 сем/курсач/Текст работы/img/trend inseparability/CiSSA.png")  # сохранение в формате PNG

plot(1:n, f_n, type = "l", lwd=3, ylim= c(-10, 10), col="red",
     xlab = "Время", ylab = "Значения ряда", main = "CiSSA разложение временного ряда")
lines(1:n, r$trend, col = "blue")
lines(1:n, r$sesonal_sin, col = "blue")
lines(1:n, r$sesonal_cos, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)


# dev.off()  # завершение сохранения










s <- ssa(f_n, L)
e <- eossa_new(s, nested.groups = list(1:30), clust_type = "distance")

g_sesonal <- grouping.auto(e, base = "eigen",
                   freq.bins = list(trend = c(1/25-eps),
                                    sesonal2 = c(1/25, 1/23),
                                    sesonal1 = c(1/13, 1/11)
                                    ),
                   threshold = 0.1)


r <- reconstruct(e, groups= g_sesonal)

plot(wcor(e, groups = 1:24), scales = list(at = c(10, 20, 30)))
Предупреждение в wcor.ossa(e, groups = 1:24) :
  Component matrices are not F-orthogonal (max F-cor is -0.529). W-cor matrix can be irrelevant

data$exp_err[1] <- mse(f_e, r$trend)  |> formatC(format = "e", digits = 1)
data$cos_err[1] <- mse(f_c, r$sesonal1) |> formatC(format = "e", digits = 1)
data$sin_err[1] <- mse(f_s, r$sesonal2) |> formatC(format = "e", digits = 1)


# png("C:/Users/nik1m/Desktop/уник/6 сем/курсач/Текст работы/img/trend inseparability/SSA.png")  # сохранение в формате PNG

plot(1:n, f_n, type = "l", lwd=3, ylim= c(-10, 10), col="red",
     xlab = "Время", ylab = "Значения ряда", main = "SSA разложение временного ряда")

lines(1:n, r$trend, type = "l", col="blue")
lines(1:n, r$sesonal1, col = "blue")
lines(1:n, r$sesonal2, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)








# Шаг 3: Преобразование данных в формат LaTeX
table_latex <- xtable(data, caption = "Example Table")

# Шаг 4: Вывод таблицы в LaTeX файл
print(table_latex, include.rownames = FALSE)
% latex table generated in R 4.2.2 by xtable 1.8-4 package
% Mon Nov  4 13:25:10 2024
\begin{table}[ht]
\centering
\begin{tabular}{llll}
  \hline
Метод & exp\_err & sin\_err & cos\_err \\ 
  \hline
SSA & 4.7e-29 & 1.1e-29 & 8.4e-30 \\ 
  CiSSA & 3.2e-02 & 2.6e-04 & 5.8e-03 \\ 
   \hline
\end{tabular}
\caption{Example Table} 
\end{table}

Данные IP

library(readxl)
Предупреждение: пакет ‘readxl’ был собран под R версии 4.2.3
data <- read_excel("Data/International_Financial_Statistics_.xlsx")
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data |> head()

Отрисовка данных IP

dates <- seq(as.Date("1970-01-01"), as.Date("2018-1-30"), by = "month")
IP_values <- data[2, -c(1, 2)] |> as.double() 
plot(dates, IP_values, type="l")

Cissa

Отрисовка трендовой составляющей чёрным цветом, основной временной ряд — красным

data_slice <- 1:537
dates_slice <- dates[data_slice]
IP_values_slice <- IP_values[data_slice]
eps <- 1/193

c <- circulant_SSA(IP_values_slice, L = 192, extend_flag = TRUE)
r <- c$t_series
r <- grouping_cissa(c,
                    groups = list(
                      trend = c(0, 1/192),
                      cycle = c(1/97, 5/95),
                      sesonal = c(1/13, 1/2+0.0001)
                    )
                    )
r_sesonal <-  grouping_cissa(c,
                             groups = list(
                              s1 = c(16/192 - eps, 16/192 + eps),
                              s2 = c(32/192 - eps, 32/192 + eps),
                              s3 = c(48/192 - eps, 48/192 + eps),
                              s4 = c(64/192 - eps, 64/192 + eps),
                              s5 = c(80/192 - eps, 80/192 + eps),
                              s6 = c(96/192 - eps, 96/192 + eps)
                             )
                             )
# cissa_trend <- r[,1] + r[,2]
# cissa_cycle <- r[, 3:11] |> rowSums()
# cissa_sesonal <- r[, c(17, 33, 49, 65, 81, 97)] |> rowSums()
# cissa_residuals <- IP_values_slice - (cissa_trend + cissa_cycle + cissa_sesonal)

cissa_trend <- r$trend
cissa_cycle <- r$cycle
cissa_sesonal <- Reduce("+", r_sesonal |> within(rm(residuals)))
cissa_residuals <- IP_values_slice - (cissa_trend + cissa_cycle + cissa_sesonal)


plot(dates_slice, IP_values_slice,
     type="l", col = "black")
lines(dates_slice, cissa_trend,
      type="l", col = "red")


plot(dates_slice, cissa_cycle,
     type="l", col = "red")


plot(dates_slice, cissa_sesonal,
     type="l", col = "red")


plot(dates_slice, cissa_residuals,
     type="l", col = "red")


plot(dates_slice, IP_values_slice,
     type="l", col = "black")
lines(dates_slice, cissa_trend+cissa_cycle+cissa_sesonal,
      type="l", col = "red")

SSA fossa

s <- ssa(IP_values_slice, L = 192)
e <- fossa(s)
# e <- eossa_new(s, nested.groups = list(1:30), clust_type = "distance")
eps <- 1/193

groups <- grouping.auto(e,
                   freq.bins = list(trend = c(1/192),
                                    cycle = c(1/97, 5/95),
                                    s1 = c(16/192 - eps, 16/192 + eps),
                                    s2 = c(32/192 - eps, 32/192 + eps),
                                    s3 = c(48/192 - eps, 48/192 + eps),
                                    s4 = c(64/192 - eps, 64/192 + eps),
                                    s5 = c(80/192 - eps, 80/192 + eps),
                                    s6 = c(96/192 - eps, 96/192 + eps)
                                    ),
                   threshold = 0)


plot(wcor(e, groups = 1:30), scales = list(at = c(10, 20, 30)),
     main = "W-correlation matrix SSA (fossa)")


r <- reconstruct(e, groups=groups)

ssa_trend_f <- r$trend
ssa_cycle_f <- r$cycle
ssa_sesonal_f <- r$s1 + r$s2 + r$s3 + r$s4 + r$s5 + r$s6
ssa_residuals_f <- IP_values_slice - (ssa_trend_f + ssa_cycle_f + ssa_sesonal_f)

plot(dates_slice, IP_values_slice,
     type="l", col = "black")
lines(dates_slice, ssa_trend_f,
      type="l", col = "magenta")


plot(dates_slice, ssa_cycle_f, 
     type="l", col = "magenta")


plot(dates_slice, ssa_sesonal_f, 
     type="l", col = "magenta")


plot(dates_slice, ssa_residuals_f,
     type="l", col = "magenta")

SSA eossa

library(Rssa)
source("eossa_new.r")
s <- ssa(IP_values_slice, L = 192)
e <- eossa_new(s, nested.groups = list(1:30), clust_type = "distance")




groups <- grouping.auto(e,
                   freq.bins = list(trend = c(1/192),
                                    cycle = c(1/97, 5/95),
                                    s1 = c(16/192 - eps, 16/192 + eps),
                                    s2 = c(32/192 - eps, 32/192 + eps),
                                    s3 = c(48/192 - eps, 48/192 + eps),
                                    s4 = c(64/192 - eps, 64/192 + eps),
                                    s5 = c(80/192 - eps, 80/192 + eps),
                                    s6 = c(96/192 - eps, 96/192 + eps)
                                    ),
                   threshold = 0)
plot(wcor(e, groups = 1:30), scales = list(at = c(10, 20, 30)),
     main = "W-correlation matrix SSA (eossa)")
Предупреждение в wcor.ossa(e, groups = 1:30) :
  Component matrices are not F-orthogonal (max F-cor is -0.0621). W-cor matrix can be irrelevant

r <- reconstruct(e, groups=groups)

ssa_trend <- r$trend
ssa_cycle <- r$cycle
ssa_sesonal <- r$s1 + r$s2 + r$s3 + r$s4 + r$s5 + r$s6
ssa_residuals <- IP_values_slice - (ssa_trend + ssa_cycle + ssa_sesonal)

plot(dates_slice, IP_values_slice,
     type="l", col = "black")
lines(dates_slice, ssa_trend,
      type="l", col = "blue")


plot(dates_slice, ssa_cycle, 
     type="l", col = "blue")


plot(dates_slice, ssa_sesonal, 
     type="l", col = "blue")


plot(dates_slice, ssa_residuals,
     type="l", col = "blue")

plot(dates_slice, IP_values_slice,
     main = "IP USA тренд",xlab = "Время", ylab = "Значение",
     type="l", col = "black")
lines(dates_slice, ssa_trend,
      type="l", col = "blue", lwd=2)
lines(dates_slice, ssa_trend_f,
      type="l", col = "magenta", lwd=2)
lines(dates_slice, cissa_trend,
      type="l", col = "red", lwd=2)
# Легенда
legend("topleft", legend = c("Весь ряд", "CiSSA тренд", "SSA тренд (eossa)", "SSA тренд (fossa)"), 
       col = c("black", "red", "blue", "magenta"), lty = 1, lwd = 3)



plot(dates_slice, ssa_cycle,
     main = "IP USA цикличность", xlab = "Время", ylab = "Значение",
     type="l", col = "blue", ylim=c(-10, 10), lwd=2)
lines(dates_slice, cissa_cycle,
      type="l", col = "red", lwd=2)
lines(dates_slice, ssa_cycle_f,
      type="l", col = "magenta", lwd=2)
# Легенда
legend("topleft", legend = c("CiSSA", "SSA (eossa)", "SSA (fossa)"), 
       col = c("red", "blue", "magenta"), lty = 1, lwd = 3)

# Настройка графиков для отображения двух графиков один под другим с общей осью X
layout(matrix(c(1, 2), nrow = 2, byrow = TRUE), heights = c(1, 1.2))

# Построение первого графика
par(mar = c(2, 4, 2, 2)) # Уменьшение нижнего отступа
plot(dates_slice, ssa_sesonal, type = "l", col = "blue", lwd = 1,
     main = "SSA (eossa) сезонность", xlab = "", ylab = "Значение")
# Добавление оси X внизу первого графика, но с пустыми метками
axis(1, labels = FALSE)

# Построение второго графика
par(mar = c(5, 4, 2, 2)) # Увеличение нижнего отступа
plot(dates_slice, ssa_sesonal_f, type = "l", col = "magenta", lwd = 1,
     main = "SSA (fossa) сезонность", xlab = "Время", ylab = "Значение")

par(mar = c(3, 4, 2, 2)) # Увеличение нижнего отступа

plot(dates_slice, cissa_sesonal, type = "l", col = "red", lwd = 1,
     main = "CiSSA сезонность", xlab = "Время", ylab = "Значение")

# Восстановление макета по умолчанию
layout(1)

NA
NA
plot(dates_slice, ssa_residuals, 
     main = "IP USA остаток", xlab = "Время", ylab = "Значение",
     type="l", col = "blue", ylim=c(-2, 2))
lines(dates_slice, cissa_residuals,
      type="l", col = "red")
lines(dates_slice, ssa_residuals_f,
      type="l", col = "magenta")
legend("topleft", legend = c("CiSSA", "SSA (eossa)", "SSA (fossa)"), 
       col = c("red", "blue", "magenta"), lty = 1, lwd = 3)

ssa_residuals |> density() |> plot()

cissa_residuals |> density() |> plot()

Отделение сигнала от шума

set.seed(100)

n_mse_tests <- function(n){
  n <- 96*2-1
  L <- 96
  sigma <- 0.1
  
  
  C <- 1
  omega_cs <- 1/12
  omega_sn <- 1/24
  a <- 1/100
  f_sum <- function(x){
    f_const(x, C = C) +
      f_cos(x, omega = omega_cs) +
      f_exp(x, a = a) +
      f_sin(x, omega = omega_sn)
  }
  
  
  f_C <- f_const |> generate_ts(n, C = C)
  f_c <- f_cos |> generate_ts(n, omega = omega_cs)
  f_s <- f_sin |> generate_ts(n, omega = omega_sn)
  f_e <- f_exp |> generate_ts(n, a = a)
  
  mse_lst <- list()
  for (i in 1:n) {
    f_noise <- rnorm(n, sd = sigma)
    
    f_n <- f_sum(1:n) + f_noise
    
    
    
    c <- circulant_SSA(f_n, L = L, extend_flag = TRUE)
    # r <- c$t_series
    r <- grouping_cissa(c, groups= list(trend = c(0, 1/1000), 
                                        sesonal2 = c(1/25, 1/23),
                                        sesonal1 = c(1/13, 1/10)
                                        ))
    
    # mse_lst$cissa <- c(mse_lst$cissa, mse(f_sum(1:n), r[, 9] + r[, 5] + r[, 1])) 
    mse_lst$cissa <- c(mse_lst$cissa,
                       mse(f_sum(1:n),
                           r$trend + r$sesonal1 + r$sesonal2)) 
    
    
    
    
    s <- ssa(f_n, L)
    # e <- eossa(s, 1:10, k = 6)
    e <- fossa(s)
    
    g_sesonal <- grouping.auto(e, base = "eigen",
                       freq.bins = list(trend = 1/1000, 
                                        sesonal2 = c(1/25, 1/23),
                                        sesonal1 = c(1/13, 1/10)
                                        ),
                       threshold = 0.5)
    
    r <- reconstruct(e, groups=c(list(exp = 1, C = 2), g_sesonal))
    
    mse_lst$ssa <- 
      c(mse_lst$ssa, mse(f_sum(1:n), r$trend + r$sesonal2 + r$sesonal1))
 
  }
  return(mse_lst)
}

res_mse_test <- n_mse_tests(10000)
# Оценка плотности
density_estimate_cissa <- density(res_mse_test$cissa)

# Построение графика плотности
plot(density_estimate_cissa, main = "Оценка плотности", 
     xlab = "Значение", ylab = "Плотность", 
     col = "blue", lwd = 2)


density_estimate_ssa <- density(res_mse_test$ssa)

# Построение графика плотности
plot(density_estimate_ssa, main = "Оценка плотности", 
     xlab = "Значение", ylab = "Плотность", 
     col = "blue", lwd = 2)


res_mse_test$cissa |> summary()
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
0.02299 0.03062 0.03345 0.03378 0.03575 0.04800 
res_mse_test$cissa |> sd()
[1] 0.004333975
res_mse_test$ssa |> summary()
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
0.000576 0.001710 0.002125 0.002228 0.002585 0.006311 
res_mse_test$ssa |> sd()
[1] 0.0008283841

Как выполняется расширение ряда

IP_values_slice |> extend(192) |> plot(type="l", lwd = 3)
c(rep(0, 192),IP_values_slice) |> lines(type="l", col="red")

Фурье преобразование

n <- 96*2
L <- 96

x <- 0:(n-1)
y1 <- cos(2 * pi / 12 * x)  # Первая компонента
y2 <- sin(2 * pi / 48 * x)  # Вторая компонента

# Создаем общий временной ряд
y <- y1 + y2

# Выполняем быстрое преобразование Фурье
fft_y <- fft(y)

# Получаем амплитуды и фазы
amplitudes <- Mod(fft_y)
phases <- Arg(fft_y)

# Индексы для частот
n <- length(y)
frequencies <- (0:(n-1)) / n

# Функция для восстановления компонент
reconstruct_fft <- function(frequencies, amplitudes, phases) {
  reconstructed <- list()
  n <- length(amplitudes)
  for (i in 1:(length(amplitudes))) {
    reconstructed[[i]] <-
      amplitudes[i] * cos(2 * pi * frequencies[i] * x + phases[i]) / n * 2
  }
  return(reconstructed)
}

# y_main <- y
# X <- hankel(y)
# K <- dim(X)[2]
# res <- list()
# for (i in 1:K){
#   y <- X[, i]
#   
#   # Выполняем быстрое преобразование Фурье
#   fft_y <- fft(y)
#   
#   # Получаем амплитуды и фазы
#   amplitudes <- Mod(fft_y)
#   phases <- Arg(fft_y)
#   
#   res[[i]] <- reconstruct_fft(frequencies, amplitudes, phases)
# }
# 
# nft <- res[[1]]
# print(nft |> length())
# res_full <- matrix(0, nrow =  |> length(), )
# 
# for (i in 1:K){
#   res_full <- 
# }

# 
# 
y_reconstructed <- reconstruct_fft(frequencies, amplitudes, phases)

# Строим графики
for (i in 1:(n)){
  plot(x, y_reconstructed[[i]], main = paste(frequencies[i]))

}


plot(x, Reduce("+", y_reconstructed), type = "l")

# lines(x, y, col = "red", type = "l", lty = 2)
x <- 1:100
y <- sin(2*pi*x)
dftmtx(10)[3, 2]
[1] 0.309017-0.9510565i
dftmtx(10)[2, 3]
[1] 0.309017-0.9510565i
(as.matrix(dftmtx(10)[3, ])) %*% t(as.matrix(Conj(dftmtx(10)[, 3])))
                      [,1]                 [,2]                 [,3]                 [,4]                 [,5]
 [1,]  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i
 [2,]  0.309017-0.9510565i  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i
 [3,] -0.809017-0.5877853i  0.309017-0.9510565i  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i
 [4,] -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i  1.000000-0.0000000i  0.309017+0.9510565i
 [5,]  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i  1.000000+0.0000000i
 [6,]  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i
 [7,]  0.309017-0.9510565i  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i
 [8,] -0.809017-0.5877853i  0.309017-0.9510565i  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i
 [9,] -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i  1.000000-0.0000000i  0.309017+0.9510565i
[10,]  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i  1.000000+0.0000000i
                      [,6]                 [,7]                 [,8]                 [,9]                [,10]
 [1,]  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i
 [2,]  0.309017-0.9510565i  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i
 [3,] -0.809017-0.5877853i  0.309017-0.9510565i  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i
 [4,] -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i  1.000000+0.0000000i  0.309017+0.9510565i
 [5,]  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i  1.000000+0.0000000i
 [6,]  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i
 [7,]  0.309017-0.9510565i  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i
 [8,] -0.809017-0.5877853i  0.309017-0.9510565i  1.000000+0.0000000i  0.309017+0.9510565i -0.809017+0.5877853i
 [9,] -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i  1.000000+0.0000000i  0.309017+0.9510565i
[10,]  0.309017+0.9510565i -0.809017+0.5877853i -0.809017-0.5877853i  0.309017-0.9510565i  1.000000+0.0000000i
dim( t(as.matrix(Conj(dftmtx(10)[3, ]))))
[1]  1 10
dim(as.matrix(dftmtx(10)[3, ]))
[1] 10  1

X <- matrix(c(32, 21, 23,
           521,631452, 25251,
           1536, 75, 38), nrow= 3)
f_mat <- dftmtx(3) / sqrt(3)
f_mat_inv <- Conj(f_mat)
X
     [,1]   [,2] [,3]
[1,]   32    521 1536
[2,]   21 631452   75
[3,]   23  25251   38
f_mat %*% f_mat_inv %*% X
      [,1]      [,2]    [,3]
[1,] 32+0i    521+0i 1536+0i
[2,] 21+0i 631452+0i   75+0i
[3,] 23+0i  25251+0i   38+0i
data_slice <- 1:538
n <- 96*2
L <- 96

x <- data_slice
y1 <- cos(2 * pi / 12 * x)  # Первая компонента
y2 <- sin(2 * pi / 48 * x)  # Вторая компонента

# Создаем общий временной ряд
y <- IP_values[data_slice]
y_extended <- y |> extend(L)
x <- 1:length(y_extended)
y <- IP_values[data_slice]

# Выполняем быстрое преобразование Фурье
fft_y <- fft(y)

# Получаем амплитуды и фазы
amplitudes <- Mod(fft_y)
phases <- Arg(fft_y)

# Индексы для частот
n <- length(y)
frequencies <- (0:(n-1)) / n

# Функция для восстановления компонент
reconstruct_fft <- function(frequencies, amplitudes, phases) {
  reconstructed <- list()
  n <- length(amplitudes)
  for (i in 1:(length(amplitudes))) {
    reconstructed[[i]] <-
      amplitudes[i] * cos(2 * pi * frequencies[i] * x + phases[i]) / n * 2
  }
  return(reconstructed)
}

y_reconstructed <- reconstruct_fft(frequencies, amplitudes, phases)

# Строим графики
for (i in 1:(n)){
  plot(x, y_reconstructed[[i]], main = paste(frequencies[i]))
  
}

data_slice <- 1:538
n <- 96*2
L <- 96

x <- data_slice
y1 <- cos(2 * pi / 12 * x)  # Первая компонента
y2 <- sin(2 * pi / 48 * x)  # Вторая компонента

# Создаем общий временной ряд
y <- IP_values[data_slice]
y_extended <- y |> extend(L)
x <- 1:length(y_extended)
y <- IP_values[data_slice]

# Выполняем быстрое преобразование Фурье
fft_y <- fft(y)

# Получаем амплитуды и фазы
amplitudes <- Mod(fft_y)
phases <- Arg(fft_y)

# Индексы для частот
n <- length(y)
frequencies <- (0:(n-1)) / n

# Функция для восстановления компонент
reconstruct_fft <- function(frequencies, amplitudes, phases) {
  reconstructed <- list()
  n <- length(amplitudes)
  for (i in 1:(length(amplitudes))) {
    reconstructed[[i]] <-
      amplitudes[i] * cos(2 * pi * frequencies[i] * x + phases[i]) / n * 2
  }
  return(reconstructed)
}

y_reconstructed <- reconstruct_fft(frequencies, amplitudes, phases)

# Строим графики
for (i in 1:(n)){
  plot(x, y_reconstructed[[i]], main = paste(frequencies[i]))
  
}

n <- 96*2-1
x <- 0:(n-1)
L <- 96
y <- sin(2*pi/12 * x)

X <- hankel(y, L)

Ft <- dftmtx(L) / sqrt(L)
Ft_inv <- t(Conj(Ft))

component_wise_mult <- function(index){
  Ft %*% t(sweep(Ft_inv, 1, X[, index], '*'))
}


averaging <- function(res_comp_wise_mult){
  K <- dim(X)[2]
  counters <- rep(0, n)
  res <- matrix(0, nrow = n, ncol = L)
  for (i in 1:K){
    res[i:(i+L-1), ] <- res[i:(i+L-1), ] + res_comp_wise_mult[[i]]
    counters[i:(i+L-1)] <- counters[i:(i+L-1)] + 1
  }
  for (i in 1:n){
    res[i, ] <- res[i, ] / counters[i]
  }
  res
}


res_comp_wise <- lapply(1:L, component_wise_mult)
print(res_comp_wise[[1]])
      [,1]                        [,2]                        [,3]                        [,4]                        [,5]
 [1,] 0+0i  1.387779e-17-4.374756e-17i  2.255141e-17+1.778092e-17i -1.561251e-17+1.994932e-17i -5.377643e-17-6.505213e-18i
 [2,] 0+0i  5.000000e-01+0.000000e+00i  6.938894e-18-6.776264e-17i  1.561251e-17-6.418477e-17i -6.938894e-18+1.170938e-17i
 [3,] 0+0i  1.040834e-17-2.569559e-17i  8.660254e-01+0.000000e+00i -3.989864e-17-3.165870e-17i -5.898060e-17-3.361027e-17i
 [4,] 0+0i -2.949030e-17+2.634611e-17i  6.938894e-17-2.667137e-17i  1.000000e+00-0.000000e+00i  2.949030e-17-1.994932e-17i
 [5,] 0+0i  5.204170e-18-4.987330e-18i -1.734723e-17-2.602085e-18i  2.949030e-17+0.000000e+00i  8.660254e-01-0.000000e+00i
 [6,] 0+0i  5.204170e-18+6.505213e-18i  0.000000e+00-3.187554e-17i -5.377643e-17-1.431147e-17i -3.469447e-18+1.144917e-16i
 [7,] 0+0i  3.469447e-18-4.770490e-18i -2.775558e-17-4.336809e-19i -3.469447e-18-6.071532e-18i  2.255141e-17-2.168404e-18i
 [8,] 0+0i -1.734723e-18+1.647987e-17i  0.000000e+00-8.673617e-18i -1.144917e-16+1.149254e-16i -6.938894e-18-1.214306e-17i
 [9,] 0+0i  0.000000e+00+6.071532e-18i  1.734723e-18-5.637851e-18i  0.000000e+00+1.734723e-18i  5.204170e-17-1.301043e-17i
[10,] 0+0i -8.673617e-19-4.336809e-19i -5.204170e-18-5.204170e-18i  2.949030e-17+5.204170e-18i  1.040834e-17+3.469447e-18i
                             [,6]                        [,7]                        [,8]                        [,9]
 [1,]  0.000000e+00+4.987330e-18i  5.777790e-34-3.274081e-33i -3.469447e-18+7.806256e-18i -2.602085e-18+0.000000e+00i
 [2,]  2.515349e-17-3.469447e-18i  3.851860e-34+1.637040e-33i  1.734723e-18+6.505213e-18i -1.040834e-17-7.806256e-18i
 [3,] -1.734723e-18+8.673617e-19i  1.213336e-32-9.629650e-35i -2.602085e-18+6.938894e-18i -1.040834e-17-2.645453e-17i
 [4,]  1.387779e-17-6.722053e-18i  0.000000e+00-2.311116e-33i -6.765422e-17+1.734723e-18i  8.673617e-18+1.734723e-18i
 [5,] -8.673617e-19+9.540979e-18i -3.851860e-34-4.188898e-33i  6.938894e-18+5.421011e-18i  0.000000e+00-7.372575e-18i
 [6,]  5.000000e-01-0.000000e+00i -9.629650e-34-2.888895e-33i -8.673617e-19-8.023096e-18i  1.214306e-17-1.691355e-17i
 [7,] -1.040834e-17-6.071532e-18i  1.224606e-16+0.000000e+00i  6.938894e-18+5.421011e-18i -1.040834e-17-8.239937e-18i
 [8,] -8.673617e-19-2.406929e-17i -7.896313e-33+9.629650e-34i -5.000000e-01-0.000000e+00i -1.214306e-17+5.811324e-17i
 [9,] -8.673617e-19+1.387779e-17i  9.629650e-34-5.488900e-33i -8.673617e-19+2.775558e-17i -8.660254e-01-0.000000e+00i
[10,] -8.673617e-19+2.385245e-18i -3.274081e-33-1.829633e-33i -2.255141e-17-4.987330e-18i -3.469447e-18+2.753874e-17i
                            [,10]                       [,11]                       [,12]                       [,13]
 [1,] -8.673617e-18+1.821460e-17i -1.734723e-18-8.673617e-19i -8.673617e-19+1.734723e-18i  0.000000e+00-3.851860e-34i
 [2,]  5.724587e-17+5.204170e-18i  7.806256e-18-1.040834e-17i  0.000000e+00+1.257675e-17i -3.851860e-34+2.311116e-33i
 [3,]  1.734723e-18-6.071532e-18i  5.030698e-17-2.775558e-17i  1.734723e-18+4.336809e-18i -3.466674e-33-3.274081e-33i
 [4,] -3.122502e-17+2.862294e-17i  6.938894e-18-6.071532e-18i  5.551115e-17+3.903128e-18i -7.703720e-34-4.237046e-33i
 [5,] -2.255141e-17+2.255141e-17i  1.387779e-17-2.211772e-17i  8.673617e-19-3.469447e-18i  2.465190e-32+9.629650e-34i
 [6,] -8.673617e-18+1.734723e-17i  2.255141e-17-3.035766e-18i  2.602085e-18+1.474515e-17i -1.540744e-33-1.155558e-33i
 [7,] -6.938894e-18+3.035766e-18i -1.144917e-16+7.372575e-18i -2.602085e-18+2.168404e-18i -5.007418e-33+1.925930e-34i
 [8,] -2.949030e-17-6.808790e-17i -8.673617e-18-2.558717e-17i -4.336809e-17+2.537033e-17i  3.466674e-33+1.540744e-33i
 [9,]  1.734723e-18+7.719519e-17i  3.469447e-18-1.257675e-17i -2.602085e-18-2.081668e-17i  1.540744e-33+2.503709e-33i
[10,] -1.000000e+00-0.000000e+00i  2.081668e-17-9.887924e-17i  1.734723e-17-9.540979e-18i  5.007418e-33+3.851860e-33i
                            [,14]                       [,15]                       [,16]                       [,17]
 [1,]  4.770490e-18-1.734723e-18i  6.938894e-18-1.734723e-18i  1.734723e-18-8.673617e-18i  1.214306e-17+3.469447e-18i
 [2,]  1.647987e-17-1.604619e-17i  2.602085e-18+6.071532e-18i -1.734723e-18+3.469447e-18i  7.806256e-18-6.071532e-18i
 [3,]  5.204170e-18+2.602085e-18i -1.561251e-17+2.428613e-17i  2.602085e-18-6.938894e-18i  0.000000e+00+9.540979e-18i
 [4,]  0.000000e+00-4.770490e-18i -1.734723e-18-1.214306e-17i  8.673617e-19-2.602085e-17i  8.673617e-19+1.734723e-18i
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 [8,] -3.469447e-18+2.775558e-17i  1.127570e-17+8.673617e-18i -2.949030e-17+2.255141e-17i
 [9,]  0.000000e+00-1.734723e-18i -1.387779e-17-8.673617e-19i  2.602085e-18+4.770490e-18i
[10,]  2.775558e-17-3.209238e-17i -8.673617e-19-1.040834e-17i -8.673617e-19-1.301043e-17i
 [ достигнута getOption("max.print") -- пропущено 86 строк ]
plot(1:L, res_comp_wise[[1]][, 2])
Предупреждение в xy.coords(x, y, xlabel, ylabel, log) :
  мнимые части убраны при преобразовании

avr <- averaging(res_comp_wise)

for (i in 1:dim(res$t_series)[2]){
  plot(x, avr[i, ])
}
Ошибка в 1:dim(res$t_series)[2] :аргумент нулевой длины
n <- 96*2-1
x <- 0:(n-1)
L <- 96
y <- sin(2*pi/12 * x)

X <- hankel(y, L)

Ft <- dftmtx(L) / sqrt(L)
Ft_inv <- t(Conj(Ft))

component_wise_mult <- function(index){
  Ft %*% t(sweep(Ft_inv, 1, X[, index], '*'))
}


averaging <- function(res_comp_wise_mult){
  K <- dim(X)[2]
  counters <- rep(0, n)
  res <- matrix(0, nrow = n, ncol = L)
  for (i in 1:K){
    res[i:(i+L-1), ] <- res[i:(i+L-1), ] + res_comp_wise_mult[[i]]
    counters[i:(i+L-1)] <- counters[i:(i+L-1)] + 1
  }
  for (i in 1:n){
    res[i, ] <- res[i, ] / counters[i]
  }
  res
}


res_comp_wise <- lapply(1:L, component_wise_mult)
print(res_comp_wise[[1]])
      [,1]                        [,2]                        [,3]                        [,4]                        [,5]
 [1,] 0+0i  1.387779e-17-4.374756e-17i  2.255141e-17+1.778092e-17i -1.561251e-17+1.994932e-17i -5.377643e-17-6.505213e-18i
 [2,] 0+0i  5.000000e-01+0.000000e+00i  6.938894e-18-6.776264e-17i  1.561251e-17-6.418477e-17i -6.938894e-18+1.170938e-17i
 [3,] 0+0i  1.040834e-17-2.569559e-17i  8.660254e-01+0.000000e+00i -3.989864e-17-3.165870e-17i -5.898060e-17-3.361027e-17i
 [4,] 0+0i -2.949030e-17+2.634611e-17i  6.938894e-17-2.667137e-17i  1.000000e+00-0.000000e+00i  2.949030e-17-1.994932e-17i
 [5,] 0+0i  5.204170e-18-4.987330e-18i -1.734723e-17-2.602085e-18i  2.949030e-17+0.000000e+00i  8.660254e-01-0.000000e+00i
 [6,] 0+0i  5.204170e-18+6.505213e-18i  0.000000e+00-3.187554e-17i -5.377643e-17-1.431147e-17i -3.469447e-18+1.144917e-16i
 [7,] 0+0i  3.469447e-18-4.770490e-18i -2.775558e-17-4.336809e-19i -3.469447e-18-6.071532e-18i  2.255141e-17-2.168404e-18i
 [8,] 0+0i -1.734723e-18+1.647987e-17i  0.000000e+00-8.673617e-18i -1.144917e-16+1.149254e-16i -6.938894e-18-1.214306e-17i
 [9,] 0+0i  0.000000e+00+6.071532e-18i  1.734723e-18-5.637851e-18i  0.000000e+00+1.734723e-18i  5.204170e-17-1.301043e-17i
[10,] 0+0i -8.673617e-19-4.336809e-19i -5.204170e-18-5.204170e-18i  2.949030e-17+5.204170e-18i  1.040834e-17+3.469447e-18i
                             [,6]                        [,7]                        [,8]                        [,9]
 [1,]  0.000000e+00+4.987330e-18i  5.777790e-34-3.274081e-33i -3.469447e-18+7.806256e-18i -2.602085e-18+0.000000e+00i
 [2,]  2.515349e-17-3.469447e-18i  3.851860e-34+1.637040e-33i  1.734723e-18+6.505213e-18i -1.040834e-17-7.806256e-18i
 [3,] -1.734723e-18+8.673617e-19i  1.213336e-32-9.629650e-35i -2.602085e-18+6.938894e-18i -1.040834e-17-2.645453e-17i
 [4,]  1.387779e-17-6.722053e-18i  0.000000e+00-2.311116e-33i -6.765422e-17+1.734723e-18i  8.673617e-18+1.734723e-18i
 [5,] -8.673617e-19+9.540979e-18i -3.851860e-34-4.188898e-33i  6.938894e-18+5.421011e-18i  0.000000e+00-7.372575e-18i
 [6,]  5.000000e-01-0.000000e+00i -9.629650e-34-2.888895e-33i -8.673617e-19-8.023096e-18i  1.214306e-17-1.691355e-17i
 [7,] -1.040834e-17-6.071532e-18i  1.224606e-16+0.000000e+00i  6.938894e-18+5.421011e-18i -1.040834e-17-8.239937e-18i
 [8,] -8.673617e-19-2.406929e-17i -7.896313e-33+9.629650e-34i -5.000000e-01-0.000000e+00i -1.214306e-17+5.811324e-17i
 [9,] -8.673617e-19+1.387779e-17i  9.629650e-34-5.488900e-33i -8.673617e-19+2.775558e-17i -8.660254e-01-0.000000e+00i
[10,] -8.673617e-19+2.385245e-18i -3.274081e-33-1.829633e-33i -2.255141e-17-4.987330e-18i -3.469447e-18+2.753874e-17i
                            [,10]                       [,11]                       [,12]                       [,13]
 [1,] -8.673617e-18+1.821460e-17i -1.734723e-18-8.673617e-19i -8.673617e-19+1.734723e-18i  0.000000e+00-3.851860e-34i
 [2,]  5.724587e-17+5.204170e-18i  7.806256e-18-1.040834e-17i  0.000000e+00+1.257675e-17i -3.851860e-34+2.311116e-33i
 [3,]  1.734723e-18-6.071532e-18i  5.030698e-17-2.775558e-17i  1.734723e-18+4.336809e-18i -3.466674e-33-3.274081e-33i
 [4,] -3.122502e-17+2.862294e-17i  6.938894e-18-6.071532e-18i  5.551115e-17+3.903128e-18i -7.703720e-34-4.237046e-33i
 [5,] -2.255141e-17+2.255141e-17i  1.387779e-17-2.211772e-17i  8.673617e-19-3.469447e-18i  2.465190e-32+9.629650e-34i
 [6,] -8.673617e-18+1.734723e-17i  2.255141e-17-3.035766e-18i  2.602085e-18+1.474515e-17i -1.540744e-33-1.155558e-33i
 [7,] -6.938894e-18+3.035766e-18i -1.144917e-16+7.372575e-18i -2.602085e-18+2.168404e-18i -5.007418e-33+1.925930e-34i
 [8,] -2.949030e-17-6.808790e-17i -8.673617e-18-2.558717e-17i -4.336809e-17+2.537033e-17i  3.466674e-33+1.540744e-33i
 [9,]  1.734723e-18+7.719519e-17i  3.469447e-18-1.257675e-17i -2.602085e-18-2.081668e-17i  1.540744e-33+2.503709e-33i
[10,] -1.000000e+00-0.000000e+00i  2.081668e-17-9.887924e-17i  1.734723e-17-9.540979e-18i  5.007418e-33+3.851860e-33i
                            [,14]                       [,15]                       [,16]                       [,17]
 [1,]  4.770490e-18-1.734723e-18i  6.938894e-18-1.734723e-18i  1.734723e-18-8.673617e-18i  1.214306e-17+3.469447e-18i
 [2,]  1.647987e-17-1.604619e-17i  2.602085e-18+6.071532e-18i -1.734723e-18+3.469447e-18i  7.806256e-18-6.071532e-18i
 [3,]  5.204170e-18+2.602085e-18i -1.561251e-17+2.428613e-17i  2.602085e-18-6.938894e-18i  0.000000e+00+9.540979e-18i
 [4,]  0.000000e+00-4.770490e-18i -1.734723e-18-1.214306e-17i  8.673617e-19-2.602085e-17i  8.673617e-19+1.734723e-18i
 [5,]  3.469447e-18-9.540979e-18i  1.734723e-18-1.734723e-18i  6.938894e-18+1.214306e-17i -4.423545e-17-3.469447e-18i
 [6,] -8.239937e-17+0.000000e+00i  1.734723e-18-2.602085e-18i -1.734723e-18-2.602085e-18i  2.602085e-18+9.540979e-18i
 [7,] -2.602085e-18-1.387779e-17i -1.422473e-16+9.540979e-18i  0.000000e+00+6.071532e-18i  1.561251e-17-1.040834e-17i
 [8,]  1.734723e-18-1.387779e-17i  6.938894e-18-1.647987e-17i -1.994932e-16+1.734723e-18i  1.734723e-18+1.647987e-17i
 [9,] -5.204170e-18-1.301043e-18i  3.469447e-18+2.732189e-17i -3.469447e-18+1.561251e-17i  5.204170e-17+0.000000e+00i
[10,]  2.688821e-17-2.818926e-18i  0.000000e+00+5.637851e-18i -2.602085e-17-7.979728e-17i  1.734723e-18-2.602085e-17i
                            [,18]                       [,19]                       [,20]                       [,21]
 [1,]  1.301043e-18+3.469447e-18i  2.118523e-33+0.000000e+00i  1.734723e-18+4.336809e-18i  1.734723e-18+1.561251e-17i
 [2,]  8.239937e-18-1.734723e-18i -5.777790e-34-1.925930e-33i  4.336809e-19+8.673617e-19i -4.336809e-18-3.469447e-18i
 [3,]  0.000000e+00-8.673617e-19i  1.117039e-32+7.703720e-34i  2.168404e-18+8.673617e-19i -8.673617e-19+0.000000e+00i
 [4,] -2.168404e-18+8.673617e-19i  1.925930e-33+0.000000e+00i -1.257675e-17-1.734723e-18i  5.204170e-18+1.040834e-17i
 [5,] -1.301043e-18+0.000000e+00i  2.311116e-33+7.703720e-34i  4.336809e-18+1.734723e-18i -1.301043e-17-2.862294e-17i
 [6,] -1.257675e-17-1.647987e-17i -3.851860e-34+4.622232e-33i  4.336809e-19+0.000000e+00i  1.734723e-18-2.602085e-18i
 [7,]  0.000000e+00+1.734723e-18i -2.465190e-32-3.851860e-34i  4.336809e-19-8.673617e-19i  1.734723e-18-2.602085e-18i
 [8,]  6.938894e-18+1.734723e-18i -1.540744e-33+3.851860e-34i -4.293441e-17+5.377643e-17i -8.673617e-19-6.071532e-18i
 [9,] -8.673617e-19+3.469447e-18i  1.540744e-33-3.851860e-34i  2.602085e-18-3.035766e-18i -5.377643e-17+5.204170e-17i
[10,] -1.387779e-17+1.301043e-17i -7.703720e-34-6.933348e-33i  8.673617e-19+1.561251e-17i -2.602085e-18-5.204170e-18i
                            [,22]                       [,23]                       [,24]                       [,25]
 [1,]  4.336809e-19+1.734723e-18i  6.071532e-18-5.204170e-18i -1.355253e-18-8.673617e-19i  0.000000e+00+0.000000e+00i
 [2,]  1.214306e-17+1.387779e-17i -2.602085e-18+0.000000e+00i -2.602085e-18-8.673617e-19i -4.814825e-35+0.000000e+00i
 [3,]  4.336809e-18-1.734723e-18i -4.163336e-17+0.000000e+00i  8.673617e-19-8.673617e-19i -3.274081e-33+0.000000e+00i
 [4,] -1.734723e-18-1.387779e-17i  1.734723e-18-1.734723e-18i -1.994932e-17+0.000000e+00i  1.925930e-34+2.311116e-33i
 [5,]  6.938894e-18+6.938894e-18i -4.336809e-19+6.938894e-18i  2.168404e-18+1.734723e-18i  0.000000e+00-2.311116e-33i
 [6,] -5.464379e-17-5.204170e-17i  0.000000e+00+6.938894e-18i  2.602085e-18-2.602085e-18i -2.503709e-33+0.000000e+00i
 [7,]  3.469447e-18-6.938894e-18i -4.336809e-17-5.724587e-17i  1.734723e-18-8.673617e-19i  3.851860e-34-2.311116e-33i
 [8,] -1.734723e-18+2.255141e-17i -2.602085e-18-1.214306e-17i -4.119968e-17-1.127570e-17i -3.851860e-34-3.081488e-33i
 [9,] -3.469447e-18+0.000000e+00i  1.734723e-18-8.673617e-19i  8.673617e-19-2.602085e-18i  0.000000e+00-7.703720e-34i
[10,] -8.847090e-17+5.464379e-17i -1.734723e-18-1.387779e-17i -4.336809e-19+1.734723e-18i -1.155558e-33+0.000000e+00i
                            [,26]                       [,27]                       [,28]                       [,29]
 [1,] -2.710505e-19+0.000000e+00i  2.385245e-18+5.204170e-18i -1.734723e-18-3.469447e-18i  1.604619e-17+1.387779e-17i
 [2,]  1.382358e-17-1.301043e-17i  6.071532e-18+6.938894e-18i  0.000000e+00-6.938894e-18i -4.336809e-19+1.734723e-18i
 [3,]  5.421011e-20-8.673617e-19i  1.431147e-17+1.734723e-18i -8.673617e-19-3.469447e-18i -6.505213e-19+0.000000e+00i
 [4,]  3.252607e-18-8.673617e-19i -1.517883e-18+0.000000e+00i -8.673617e-19+1.734723e-18i  8.673617e-19-1.734723e-18i
 [5,] -8.673617e-19+2.602085e-18i  1.084202e-18+0.000000e+00i  2.168404e-18-1.734723e-18i -2.775558e-17-2.949030e-17i
 [6,]  6.938894e-18+0.000000e+00i -4.336809e-19+1.734723e-18i -5.637851e-18+0.000000e+00i -1.301043e-18-1.734723e-18i
 [7,]  4.553649e-18+3.469447e-18i  2.862294e-17-3.989864e-17i  2.602085e-18-1.734723e-18i  1.734723e-18+0.000000e+00i
 [8,]  8.673617e-19+0.000000e+00i  2.602085e-18+1.734723e-18i  5.421011e-17+1.387779e-17i  3.469447e-18+3.469447e-18i
 [9,]  4.336809e-19+1.734723e-18i -1.301043e-18+1.734723e-18i -3.469447e-18-1.734723e-18i  4.336809e-19+1.214306e-17i
[10,]  4.336809e-19-2.602085e-18i -1.734723e-18-3.469447e-18i  8.673617e-19+5.204170e-18i  6.505213e-18+5.204170e-18i
                            [,30]                       [,31]                       [,32]                       [,33]
 [1,]  6.505213e-19+8.673617e-19i  0.000000e+00+0.000000e+00i  1.301043e-18-8.673617e-19i -2.688821e-17+0.000000e+00i
 [2,]  7.155734e-18+8.673617e-19i  4.622232e-33-3.081488e-33i -4.336809e-19-8.673617e-19i -3.469447e-18+1.734723e-18i
 [3,]  8.673617e-19+8.673617e-19i -2.619265e-32+0.000000e+00i  1.301043e-18+2.602085e-18i -6.938894e-18-1.734723e-18i
 [4,]  1.084202e-18+0.000000e+00i  4.622232e-33+0.000000e+00i -6.505213e-19+7.806256e-18i  1.301043e-18+3.469447e-18i
 [5,] -1.084202e-18+0.000000e+00i -2.696302e-32-1.540744e-32i  8.673617e-19+3.469447e-18i -1.474515e-17+0.000000e+00i
 [6,] -5.854692e-18-2.168404e-17i -1.540744e-33+6.162976e-33i -2.168404e-19+0.000000e+00i -1.301043e-18+1.734723e-18i
 [7,] -1.517883e-18-1.734723e-18i  2.187856e-31-4.930381e-32i -6.505213e-19-8.673617e-19i  5.637851e-18-1.734723e-18i
 [8,] -3.035766e-18+3.469447e-18i -6.162976e-33+0.000000e+00i -6.505213e-18+2.081668e-17i -3.469447e-18-3.469447e-18i
 [9,]  1.517883e-18+2.602085e-18i -2.465190e-32+3.081488e-33i  0.000000e+00+8.673617e-19i -6.722053e-17-2.775558e-17i
[10,] -6.505213e-19-6.071532e-18i -1.386670e-32-1.848893e-32i  2.168404e-19-3.469447e-18i -1.734723e-18-3.469447e-18i
                            [,34]                       [,35]                       [,36]                       [,37]
 [1,] -1.734723e-18+5.204170e-18i  4.336809e-18+0.000000e+00i  8.673617e-19-8.673617e-19i  2.311116e-33+0.000000e+00i
 [2,]  0.000000e+00+1.561251e-17i -8.673617e-19-8.673617e-19i  0.000000e+00+0.000000e+00i  3.081488e-33+7.703720e-34i
 [3,]  6.071532e-18+1.734723e-18i -1.474515e-17-3.989864e-17i -8.673617e-19-8.673617e-19i -7.703720e-34+0.000000e+00i
 [4,]  1.734723e-18+0.000000e+00i  1.734723e-18+0.000000e+00i  1.431147e-17+0.000000e+00i  1.540744e-33+0.000000e+00i
 [5,] -8.673617e-19+3.469447e-18i -1.734723e-18-1.734723e-18i -4.336809e-19-8.673617e-19i -2.465190e-32-1.232595e-32i
 [6,]  2.515349e-17+2.602085e-17i -8.673617e-19+0.000000e+00i -2.602085e-18+0.000000e+00i -1.155558e-33+7.703720e-34i
 [7,]  7.372575e-18+3.469447e-18i -8.673617e-19+1.387779e-17i  2.818926e-18+8.673617e-19i -5.777790e-33+0.000000e+00i
 [8,]  1.301043e-18+6.938894e-18i  3.469447e-18+3.469447e-18i -2.168404e-17+6.938894e-18i -7.703720e-34+0.000000e+00i
 [9,]  8.673617e-19+3.469447e-18i  0.000000e+00-1.734723e-18i -1.734723e-18+8.673617e-19i  3.851860e-34+0.000000e+00i
[10,] -4.336809e-17-4.336809e-17i  5.204170e-18+6.938894e-18i  3.035766e-18+0.000000e+00i -1.155558e-33-3.081488e-33i
                            [,38]                       [,39]                       [,40]                       [,41]
 [1,] -8.673617e-19+0.000000e+00i -6.938894e-18-8.673617e-19i -1.734723e-18+1.734723e-18i  6.938894e-18-6.071532e-18i
 [2,]  1.431147e-17-3.903128e-18i  0.000000e+00+0.000000e+00i  1.734723e-18+0.000000e+00i -1.734723e-18+0.000000e+00i
 [3,]  0.000000e+00+8.673617e-19i -2.688821e-17+0.000000e+00i -1.734723e-18+0.000000e+00i  4.336809e-18+3.469447e-18i
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 [5,]  4.336809e-19+0.000000e+00i -1.232595e-32+1.232595e-32i -2.168404e-19+3.469447e-18i -1.387779e-17+1.387779e-17i
 [6,] -6.288373e-18+1.474515e-17i  1.232595e-32-2.465190e-32i -1.084202e-18+8.673617e-19i  3.903128e-18-3.469447e-18i
 [7,] -2.602085e-18+1.734723e-18i  1.910523e-31-3.451266e-31i -2.818926e-18+1.734723e-18i  0.000000e+00+0.000000e+00i
 [8,]  4.336809e-19+8.673617e-19i  1.540744e-32+0.000000e+00i -4.336809e-19+4.857226e-17i -1.734723e-18+1.734723e-18i
 [9,]  1.951564e-18-1.734723e-18i -4.622232e-32+6.162976e-33i -8.673617e-19+8.673617e-19i  4.076600e-17+1.127570e-16i
[10,]  1.387779e-17+6.071532e-18i -3.081488e-33-1.232595e-32i -4.336809e-19+2.602085e-18i -8.673617e-19+0.000000e+00i
                            [,82]                       [,83]                       [,84]                       [,85]
 [1,]  3.469447e-18+5.204170e-18i  8.673617e-19+8.673617e-19i -8.673617e-19+0.000000e+00i  1.540744e-33+0.000000e+00i
 [2,]  1.908196e-17+2.255141e-17i -1.734723e-18+1.734723e-18i -2.168404e-18-8.673617e-19i  3.081488e-33+1.078521e-32i
 [3,] -2.602085e-18+3.469447e-18i -1.734723e-18+2.255141e-17i -1.734723e-18+8.673617e-19i -3.081488e-33+5.238529e-32i
 [4,]  1.734723e-18-3.469447e-18i  7.806256e-18+1.734723e-18i  1.734723e-18-1.127570e-17i  4.622232e-33+2.157042e-32i
 [5,]  8.673617e-19-1.734723e-18i  5.204170e-18+0.000000e+00i  1.734723e-18+8.673617e-19i  1.032298e-31-6.162976e-33i
 [6,] -1.301043e-17+2.775558e-17i -2.602085e-18-1.734723e-18i  0.000000e+00-6.938894e-18i  0.000000e+00-3.081488e-33i
 [7,] -4.336809e-19+3.469447e-18i -1.387779e-17+1.734723e-18i  2.168404e-19-2.602085e-18i -9.244464e-33+0.000000e+00i
 [8,]  5.637851e-18+0.000000e+00i -2.602085e-18+0.000000e+00i -7.155734e-18+6.938894e-18i -2.311116e-33-3.081488e-33i
 [9,]  2.602085e-18-1.734723e-18i  0.000000e+00-1.734723e-18i  1.734723e-18+0.000000e+00i -3.851860e-33+0.000000e+00i
[10,]  2.949030e-17+6.765422e-17i  8.673617e-19-3.469447e-18i  4.336809e-19-8.673617e-19i  1.540744e-33-9.244464e-33i
                            [,86]                       [,87]                       [,88]                       [,89]
 [1,] -1.734723e-18-2.602085e-18i  2.602085e-18+8.673617e-19i -1.734723e-18-1.561251e-17i  5.030698e-17+5.290907e-17i
 [2,]  1.431147e-17+1.474515e-17i -8.673617e-19+5.204170e-18i  0.000000e+00+1.040834e-17i  4.336809e-18+8.673617e-19i
 [3,] -8.673617e-19-1.734723e-18i -1.387779e-17-2.515349e-17i  1.734723e-18+3.469447e-18i  3.469447e-18-1.214306e-17i
 [4,] -1.344411e-17-1.734723e-18i  5.204170e-18-6.071532e-18i  2.168404e-17+2.688821e-17i  2.602085e-18+8.673617e-18i
 [5,]  3.035766e-18+0.000000e+00i -1.734723e-18-1.734723e-18i -5.204170e-18-1.734723e-18i  5.464379e-17+2.775558e-17i
 [6,] -2.818926e-17+2.602085e-17i  6.938894e-18-6.071532e-18i -8.673617e-19-1.040834e-17i -2.602085e-18-1.734723e-18i
 [7,] -1.301043e-18-6.071532e-18i -5.377643e-17+2.862294e-17i -8.673617e-19-1.734723e-18i  8.673617e-19-1.127570e-17i
 [8,] -2.602085e-18+2.602085e-18i  8.673617e-19-3.469447e-18i -7.112366e-17+5.724587e-17i  5.204170e-18+1.040834e-17i
 [9,]  1.084202e-18-1.734723e-18i  8.673617e-19+0.000000e+00i -8.673617e-19+3.469447e-18i  1.474515e-17-2.602085e-17i
[10,] -2.168404e-19+6.071532e-18i -4.336809e-19-3.469447e-18i  2.602085e-18-3.469447e-18i  6.071532e-18-1.734723e-18i
                            [,90]                       [,91]                       [,92]                       [,93]
 [1,] -6.071532e-18+5.204170e-18i -4.314083e-32-3.081488e-32i  8.673617e-18-2.385245e-18i -1.214306e-17+3.035766e-18i
 [2,] -1.214306e-17+1.301043e-18i -8.628166e-32-3.081488e-32i  1.734723e-18-2.168404e-18i  2.081668e-17+1.734723e-18i
 [3,]  1.734723e-18-7.806256e-18i -2.095412e-31+1.848893e-32i  0.000000e+00+8.673617e-19i  1.040834e-17-3.079134e-17i
 [4,] -6.071532e-18-6.938894e-18i  1.232595e-32-1.848893e-32i  0.000000e+00-8.673617e-19i  1.734723e-18-8.673617e-19i
 [5,]  4.336809e-18-4.336809e-19i -4.930381e-32+1.047706e-31i -1.734723e-18-4.336809e-18i -5.637851e-17+2.775558e-17i
 [6,]  4.119968e-17+2.775558e-17i  1.232595e-32+1.232595e-32i  6.938894e-18-9.107298e-18i -1.734723e-18+3.469447e-18i
 [7,]  2.602085e-18-2.602085e-18i  5.916457e-31+1.232595e-32i -8.673617e-19-1.734723e-18i  6.938894e-18+0.000000e+00i
 [8,] -2.602085e-18-6.938894e-18i  1.848893e-32-6.779273e-32i  3.989864e-17+2.515349e-17i -1.734723e-18-8.673617e-19i
 [9,]  3.035766e-18-8.673617e-19i -5.546678e-32+9.860761e-32i  4.336809e-18+0.000000e+00i  6.852158e-17+1.734723e-18i
[10,]  4.336809e-19-8.673617e-19i  0.000000e+00+6.162976e-33i  2.602085e-18+5.204170e-18i  1.734723e-18+7.806256e-18i
                            [,94]                       [,95]                       [,96]
 [1,] -1.040834e-17+2.168404e-18i  3.122502e-17+7.372575e-18i -1.561251e-17+5.009014e-17i
 [2,] -5.724587e-17-1.205633e-16i  8.673617e-18+4.163336e-17i -1.734723e-18-2.688821e-17i
 [3,] -2.081668e-17+2.602085e-18i  5.551115e-17-5.637851e-17i -1.734723e-18+1.214306e-17i
 [4,] -6.938894e-18+5.030698e-17i  0.000000e+00+1.040834e-17i  2.862294e-17-2.797242e-17i
 [5,] -6.938894e-18+3.469447e-18i  1.734723e-17+2.341877e-17i  8.673617e-19+5.637851e-18i
 [6,] -1.092876e-16+4.336809e-18i -1.734723e-18-1.040834e-17i  8.673617e-19-1.734723e-18i
 [7,]  3.469447e-18-1.561251e-17i -2.428613e-17+5.030698e-17i  3.469447e-18-3.903128e-18i
 [8,] -3.469447e-18+2.775558e-17i  1.127570e-17+8.673617e-18i -2.949030e-17+2.255141e-17i
 [9,]  0.000000e+00-1.734723e-18i -1.387779e-17-8.673617e-19i  2.602085e-18+4.770490e-18i
[10,]  2.775558e-17-3.209238e-17i -8.673617e-19-1.040834e-17i -8.673617e-19-1.301043e-17i
 [ достигнута getOption("max.print") -- пропущено 86 строк ]
plot(1:L, res_comp_wise[[1]][, 2])
Предупреждение в xy.coords(x, y, xlabel, ylabel, log) :
  мнимые части убраны при преобразовании

avr <- averaging(res_comp_wise)

for (i in 1:dim(res$t_series)[2]){
  plot(x, avr[i, ])
}
Ошибка в 1:dim(res$t_series)[2] :аргумент нулевой длины

CiSSA через Фурье

n <- 96*2-1
x <- 0:(n-1)
L <- 96
y <- sin(2*pi/12 * x)

X <- hankel(y, L)

Ft <- dftmtx(L) / sqrt(L)
Ft_inv <- t(Conj(Ft))

component_wise_mult <- function(index){
  Ft %*% t(sweep(Ft_inv, 1, X[, index], '*'))
}


averaging <- function(res_comp_wise_mult){
  K <- dim(X)[2]
  counters <- rep(0, n)
  res <- matrix(0, nrow = n, ncol = L)
  for (i in 1:K){
    res[i:(i+L-1), ] <- res[i:(i+L-1), ] + res_comp_wise_mult[[i]]
    counters[i:(i+L-1)] <- counters[i:(i+L-1)] + 1
  }
  for (i in 1:n){
    res[i, ] <- res[i, ] / counters[i]
  }
  res
}


res_comp_wise <- lapply(1:L, component_wise_mult)
print(res_comp_wise[[1]])
      [,1]                        [,2]                        [,3]                        [,4]                        [,5]
 [1,] 0+0i  1.387779e-17-4.374756e-17i  2.255141e-17+1.778092e-17i -1.561251e-17+1.994932e-17i -5.377643e-17-6.505213e-18i
 [2,] 0+0i  5.000000e-01+0.000000e+00i  6.938894e-18-6.776264e-17i  1.561251e-17-6.418477e-17i -6.938894e-18+1.170938e-17i
 [3,] 0+0i  1.040834e-17-2.569559e-17i  8.660254e-01+0.000000e+00i -3.989864e-17-3.165870e-17i -5.898060e-17-3.361027e-17i
 [4,] 0+0i -2.949030e-17+2.634611e-17i  6.938894e-17-2.667137e-17i  1.000000e+00-0.000000e+00i  2.949030e-17-1.994932e-17i
 [5,] 0+0i  5.204170e-18-4.987330e-18i -1.734723e-17-2.602085e-18i  2.949030e-17+0.000000e+00i  8.660254e-01-0.000000e+00i
 [6,] 0+0i  5.204170e-18+6.505213e-18i  0.000000e+00-3.187554e-17i -5.377643e-17-1.431147e-17i -3.469447e-18+1.144917e-16i
 [7,] 0+0i  3.469447e-18-4.770490e-18i -2.775558e-17-4.336809e-19i -3.469447e-18-6.071532e-18i  2.255141e-17-2.168404e-18i
 [8,] 0+0i -1.734723e-18+1.647987e-17i  0.000000e+00-8.673617e-18i -1.144917e-16+1.149254e-16i -6.938894e-18-1.214306e-17i
 [9,] 0+0i  0.000000e+00+6.071532e-18i  1.734723e-18-5.637851e-18i  0.000000e+00+1.734723e-18i  5.204170e-17-1.301043e-17i
[10,] 0+0i -8.673617e-19-4.336809e-19i -5.204170e-18-5.204170e-18i  2.949030e-17+5.204170e-18i  1.040834e-17+3.469447e-18i
                             [,6]                        [,7]                        [,8]                        [,9]
 [1,]  0.000000e+00+4.987330e-18i  5.777790e-34-3.274081e-33i -3.469447e-18+7.806256e-18i -2.602085e-18+0.000000e+00i
 [2,]  2.515349e-17-3.469447e-18i  3.851860e-34+1.637040e-33i  1.734723e-18+6.505213e-18i -1.040834e-17-7.806256e-18i
 [3,] -1.734723e-18+8.673617e-19i  1.213336e-32-9.629650e-35i -2.602085e-18+6.938894e-18i -1.040834e-17-2.645453e-17i
 [4,]  1.387779e-17-6.722053e-18i  0.000000e+00-2.311116e-33i -6.765422e-17+1.734723e-18i  8.673617e-18+1.734723e-18i
 [5,] -8.673617e-19+9.540979e-18i -3.851860e-34-4.188898e-33i  6.938894e-18+5.421011e-18i  0.000000e+00-7.372575e-18i
 [6,]  5.000000e-01-0.000000e+00i -9.629650e-34-2.888895e-33i -8.673617e-19-8.023096e-18i  1.214306e-17-1.691355e-17i
 [7,] -1.040834e-17-6.071532e-18i  1.224606e-16+0.000000e+00i  6.938894e-18+5.421011e-18i -1.040834e-17-8.239937e-18i
 [8,] -8.673617e-19-2.406929e-17i -7.896313e-33+9.629650e-34i -5.000000e-01-0.000000e+00i -1.214306e-17+5.811324e-17i
 [9,] -8.673617e-19+1.387779e-17i  9.629650e-34-5.488900e-33i -8.673617e-19+2.775558e-17i -8.660254e-01-0.000000e+00i
[10,] -8.673617e-19+2.385245e-18i -3.274081e-33-1.829633e-33i -2.255141e-17-4.987330e-18i -3.469447e-18+2.753874e-17i
                            [,10]                       [,11]                       [,12]                       [,13]
 [1,] -8.673617e-18+1.821460e-17i -1.734723e-18-8.673617e-19i -8.673617e-19+1.734723e-18i  0.000000e+00-3.851860e-34i
 [2,]  5.724587e-17+5.204170e-18i  7.806256e-18-1.040834e-17i  0.000000e+00+1.257675e-17i -3.851860e-34+2.311116e-33i
 [3,]  1.734723e-18-6.071532e-18i  5.030698e-17-2.775558e-17i  1.734723e-18+4.336809e-18i -3.466674e-33-3.274081e-33i
 [4,] -3.122502e-17+2.862294e-17i  6.938894e-18-6.071532e-18i  5.551115e-17+3.903128e-18i -7.703720e-34-4.237046e-33i
 [5,] -2.255141e-17+2.255141e-17i  1.387779e-17-2.211772e-17i  8.673617e-19-3.469447e-18i  2.465190e-32+9.629650e-34i
 [6,] -8.673617e-18+1.734723e-17i  2.255141e-17-3.035766e-18i  2.602085e-18+1.474515e-17i -1.540744e-33-1.155558e-33i
 [7,] -6.938894e-18+3.035766e-18i -1.144917e-16+7.372575e-18i -2.602085e-18+2.168404e-18i -5.007418e-33+1.925930e-34i
 [8,] -2.949030e-17-6.808790e-17i -8.673617e-18-2.558717e-17i -4.336809e-17+2.537033e-17i  3.466674e-33+1.540744e-33i
 [9,]  1.734723e-18+7.719519e-17i  3.469447e-18-1.257675e-17i -2.602085e-18-2.081668e-17i  1.540744e-33+2.503709e-33i
[10,] -1.000000e+00-0.000000e+00i  2.081668e-17-9.887924e-17i  1.734723e-17-9.540979e-18i  5.007418e-33+3.851860e-33i
                            [,14]                       [,15]                       [,16]                       [,17]
 [1,]  4.770490e-18-1.734723e-18i  6.938894e-18-1.734723e-18i  1.734723e-18-8.673617e-18i  1.214306e-17+3.469447e-18i
 [2,]  1.647987e-17-1.604619e-17i  2.602085e-18+6.071532e-18i -1.734723e-18+3.469447e-18i  7.806256e-18-6.071532e-18i
 [3,]  5.204170e-18+2.602085e-18i -1.561251e-17+2.428613e-17i  2.602085e-18-6.938894e-18i  0.000000e+00+9.540979e-18i
 [4,]  0.000000e+00-4.770490e-18i -1.734723e-18-1.214306e-17i  8.673617e-19-2.602085e-17i  8.673617e-19+1.734723e-18i
 [5,]  3.469447e-18-9.540979e-18i  1.734723e-18-1.734723e-18i  6.938894e-18+1.214306e-17i -4.423545e-17-3.469447e-18i
 [6,] -8.239937e-17+0.000000e+00i  1.734723e-18-2.602085e-18i -1.734723e-18-2.602085e-18i  2.602085e-18+9.540979e-18i
 [7,] -2.602085e-18-1.387779e-17i -1.422473e-16+9.540979e-18i  0.000000e+00+6.071532e-18i  1.561251e-17-1.040834e-17i
 [8,]  1.734723e-18-1.387779e-17i  6.938894e-18-1.647987e-17i -1.994932e-16+1.734723e-18i  1.734723e-18+1.647987e-17i
 [9,] -5.204170e-18-1.301043e-18i  3.469447e-18+2.732189e-17i -3.469447e-18+1.561251e-17i  5.204170e-17+0.000000e+00i
[10,]  2.688821e-17-2.818926e-18i  0.000000e+00+5.637851e-18i -2.602085e-17-7.979728e-17i  1.734723e-18-2.602085e-17i
                            [,18]                       [,19]                       [,20]                       [,21]
 [1,]  1.301043e-18+3.469447e-18i  2.118523e-33+0.000000e+00i  1.734723e-18+4.336809e-18i  1.734723e-18+1.561251e-17i
 [2,]  8.239937e-18-1.734723e-18i -5.777790e-34-1.925930e-33i  4.336809e-19+8.673617e-19i -4.336809e-18-3.469447e-18i
 [3,]  0.000000e+00-8.673617e-19i  1.117039e-32+7.703720e-34i  2.168404e-18+8.673617e-19i -8.673617e-19+0.000000e+00i
 [4,] -2.168404e-18+8.673617e-19i  1.925930e-33+0.000000e+00i -1.257675e-17-1.734723e-18i  5.204170e-18+1.040834e-17i
 [5,] -1.301043e-18+0.000000e+00i  2.311116e-33+7.703720e-34i  4.336809e-18+1.734723e-18i -1.301043e-17-2.862294e-17i
 [6,] -1.257675e-17-1.647987e-17i -3.851860e-34+4.622232e-33i  4.336809e-19+0.000000e+00i  1.734723e-18-2.602085e-18i
 [7,]  0.000000e+00+1.734723e-18i -2.465190e-32-3.851860e-34i  4.336809e-19-8.673617e-19i  1.734723e-18-2.602085e-18i
 [8,]  6.938894e-18+1.734723e-18i -1.540744e-33+3.851860e-34i -4.293441e-17+5.377643e-17i -8.673617e-19-6.071532e-18i
 [9,] -8.673617e-19+3.469447e-18i  1.540744e-33-3.851860e-34i  2.602085e-18-3.035766e-18i -5.377643e-17+5.204170e-17i
[10,] -1.387779e-17+1.301043e-17i -7.703720e-34-6.933348e-33i  8.673617e-19+1.561251e-17i -2.602085e-18-5.204170e-18i
                            [,22]                       [,23]                       [,24]                       [,25]
 [1,]  4.336809e-19+1.734723e-18i  6.071532e-18-5.204170e-18i -1.355253e-18-8.673617e-19i  0.000000e+00+0.000000e+00i
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 [5,] -8.673617e-19+0.000000e+00i  8.673617e-19+0.000000e+00i  1.734723e-18+0.000000e+00i -1.734723e-18+6.938894e-18i
 [6,]  2.081668e-17-1.040834e-17i  0.000000e+00+0.000000e+00i  0.000000e+00+8.673617e-19i -8.673617e-19+0.000000e+00i
 [7,]  8.673617e-19+4.336809e-19i  1.214306e-17-2.081668e-17i  1.734723e-18-8.673617e-19i  7.806256e-18-8.673617e-19i
 [8,]  0.000000e+00+0.000000e+00i -3.469447e-18-8.673617e-19i  1.561251e-17+6.938894e-18i -1.734723e-18+0.000000e+00i
 [9,]  0.000000e+00-6.505213e-19i  0.000000e+00+4.336809e-19i  1.734723e-18-8.673617e-19i -1.734723e-18+6.071532e-18i
[10,]  6.938894e-18+6.938894e-18i  0.000000e+00-8.673617e-19i  1.734723e-18+0.000000e+00i  3.469447e-18+8.673617e-19i
                            [,66]                       [,67]                       [,68]                       [,69]
 [1,] -4.336809e-19-8.673617e-19i  6.162976e-33+1.848893e-32i  6.505213e-19+0.000000e+00i  4.336809e-19-1.214306e-17i
 [2,]  4.336809e-19-6.071532e-18i -9.244464e-33+1.232595e-32i  4.770490e-18-8.673617e-19i  8.673617e-19+0.000000e+00i
 [3,]  1.301043e-18-1.734723e-18i -9.860761e-32+9.860761e-32i  1.301043e-18+0.000000e+00i -4.336809e-19-1.734723e-18i
 [4,] -3.469447e-18+3.469447e-18i  1.232595e-32-3.081488e-32i -1.951564e-17+6.071532e-18i  3.469447e-18-3.469447e-18i
 [5,] -1.734723e-18+0.000000e+00i  2.465190e-32-6.162976e-33i  3.903128e-18-2.602085e-18i  2.775558e-17-1.387779e-17i
 [6,] -5.637851e-18+6.938894e-18i  6.162976e-33-1.232595e-32i  3.035766e-18+0.000000e+00i  0.000000e+00-8.673617e-19i
 [7,]  1.734723e-18-4.336809e-19i -4.930381e-32+4.314083e-32i -4.336809e-19-8.673617e-19i  4.336809e-18+0.000000e+00i
 [8,] -8.673617e-19+0.000000e+00i  0.000000e+00-1.232595e-32i  1.301043e-18+6.938894e-18i  0.000000e+00+8.673617e-19i
 [9,]  0.000000e+00+0.000000e+00i  6.162976e-33-6.162976e-33i -8.673617e-19+0.000000e+00i  1.561251e-17+0.000000e+00i
[10,]  7.806256e-18+1.734723e-17i -1.232595e-32+0.000000e+00i  3.469447e-18-4.336809e-19i  1.734723e-18-8.673617e-19i
                            [,70]                       [,71]                       [,72]                       [,73]
 [1,]  2.168404e-18+3.469447e-18i -6.938894e-18+1.734723e-18i  3.252607e-19+0.000000e+00i  0.000000e+00+0.000000e+00i
 [2,]  8.673617e-19-2.775558e-17i -1.301043e-18+3.469447e-18i -2.168404e-19+0.000000e+00i -6.740755e-33+3.081488e-33i
 [3,]  2.602085e-18+3.469447e-18i -1.214306e-17-6.938894e-18i -4.336809e-19-8.673617e-19i -3.851860e-34+3.081488e-33i
 [4,]  1.734723e-18-5.204170e-18i -3.903128e-18+0.000000e+00i -5.637851e-18-6.938894e-18i  2.696302e-33+0.000000e+00i
 [5,] -1.734723e-18-1.734723e-18i  6.071532e-18-1.734723e-18i -4.336809e-19+8.673617e-19i -2.157042e-32+0.000000e+00i
 [6,]  1.474515e-17-2.775558e-17i  0.000000e+00+0.000000e+00i  4.336809e-19-8.673617e-19i -6.933348e-33+3.081488e-33i
 [7,]  6.938894e-18-5.204170e-18i -4.163336e-17+1.387779e-17i -4.336809e-19-8.673617e-19i  8.474092e-33+0.000000e+00i
 [8,] -1.734723e-18+0.000000e+00i  0.000000e+00-8.673617e-19i  1.387779e-17-1.301043e-17i  3.081488e-33+0.000000e+00i
 [9,]  3.469447e-18-3.469447e-18i  0.000000e+00+0.000000e+00i -1.734723e-18+8.673617e-19i -2.465190e-32+1.232595e-32i
[10,]  0.000000e+00+1.040834e-17i -1.734723e-18+8.673617e-19i  0.000000e+00+0.000000e+00i  1.540744e-33+0.000000e+00i
                            [,74]                       [,75]                       [,76]                       [,77]
 [1,]  1.084202e-18-8.673617e-19i -4.770490e-18+0.000000e+00i -1.734723e-18-1.734723e-18i -1.344411e-17-2.775558e-17i
 [2,] -6.830474e-18+0.000000e+00i -1.192622e-18+1.734723e-18i -1.734723e-18-6.938894e-18i -3.035766e-18+1.734723e-18i
 [3,] -7.589415e-19-8.673617e-19i  4.141652e-17+1.387779e-17i  8.673617e-19-3.469447e-18i  7.372575e-18-1.734723e-18i
 [4,]  2.168404e-19+3.469447e-18i  2.168404e-18-1.734723e-18i  2.862294e-17+1.734723e-18i -2.168404e-19+0.000000e+00i
 [5,]  8.673617e-19-8.673617e-19i  4.770490e-18-1.734723e-18i -5.637851e-18+1.734723e-18i  4.293441e-17+1.734723e-18i
 [6,]  1.973248e-17+6.071532e-18i  6.505213e-19+0.000000e+00i -6.505213e-18-1.734723e-18i -1.734723e-18+0.000000e+00i
 [7,]  2.602085e-18-2.602085e-18i  1.301043e-18-1.734723e-18i -2.602085e-18+0.000000e+00i -2.168404e-18-6.938894e-18i
 [8,] -8.673617e-19+0.000000e+00i  4.770490e-18-3.469447e-18i -1.431147e-17-1.734723e-18i -1.301043e-18+1.734723e-18i
 [9,] -1.734723e-18+1.734723e-18i  4.336809e-18+0.000000e+00i  0.000000e+00+3.469447e-18i  1.387779e-17-1.387779e-17i
[10,] -6.505213e-18+8.673617e-19i -8.673617e-19-1.734723e-18i  8.673617e-19+1.734723e-18i  1.734723e-18-1.734723e-18i
                            [,78]                       [,79]                       [,80]                       [,81]
 [1,] -1.517883e-18+8.673617e-19i  4.622232e-32+0.000000e+00i -2.602085e-18+1.734723e-18i  8.673617e-19+2.602085e-18i
 [2,]  1.387779e-17-8.673617e-19i  5.238529e-32-4.930381e-32i  2.602085e-18+1.734723e-18i -2.602085e-18-3.469447e-18i
 [3,]  4.336809e-19+2.602085e-18i  3.081488e-33+1.047706e-31i -8.673617e-19+3.469447e-18i -5.204170e-18-1.734723e-18i
 [4,]  1.192622e-18+0.000000e+00i -3.389637e-32+2.465190e-32i -4.336809e-19+2.168404e-17i  6.071532e-18-3.469447e-18i
 [5,]  4.336809e-19+0.000000e+00i -1.232595e-32+1.232595e-32i -2.168404e-19+3.469447e-18i -1.387779e-17+1.387779e-17i
 [6,] -6.288373e-18+1.474515e-17i  1.232595e-32-2.465190e-32i -1.084202e-18+8.673617e-19i  3.903128e-18-3.469447e-18i
 [7,] -2.602085e-18+1.734723e-18i  1.910523e-31-3.451266e-31i -2.818926e-18+1.734723e-18i  0.000000e+00+0.000000e+00i
 [8,]  4.336809e-19+8.673617e-19i  1.540744e-32+0.000000e+00i -4.336809e-19+4.857226e-17i -1.734723e-18+1.734723e-18i
 [9,]  1.951564e-18-1.734723e-18i -4.622232e-32+6.162976e-33i -8.673617e-19+8.673617e-19i  4.076600e-17+1.127570e-16i
[10,]  1.387779e-17+6.071532e-18i -3.081488e-33-1.232595e-32i -4.336809e-19+2.602085e-18i -8.673617e-19+0.000000e+00i
                            [,82]                       [,83]                       [,84]                       [,85]
 [1,]  3.469447e-18+5.204170e-18i  8.673617e-19+8.673617e-19i -8.673617e-19+0.000000e+00i  1.540744e-33+0.000000e+00i
 [2,]  1.908196e-17+2.255141e-17i -1.734723e-18+1.734723e-18i -2.168404e-18-8.673617e-19i  3.081488e-33+1.078521e-32i
 [3,] -2.602085e-18+3.469447e-18i -1.734723e-18+2.255141e-17i -1.734723e-18+8.673617e-19i -3.081488e-33+5.238529e-32i
 [4,]  1.734723e-18-3.469447e-18i  7.806256e-18+1.734723e-18i  1.734723e-18-1.127570e-17i  4.622232e-33+2.157042e-32i
 [5,]  8.673617e-19-1.734723e-18i  5.204170e-18+0.000000e+00i  1.734723e-18+8.673617e-19i  1.032298e-31-6.162976e-33i
 [6,] -1.301043e-17+2.775558e-17i -2.602085e-18-1.734723e-18i  0.000000e+00-6.938894e-18i  0.000000e+00-3.081488e-33i
 [7,] -4.336809e-19+3.469447e-18i -1.387779e-17+1.734723e-18i  2.168404e-19-2.602085e-18i -9.244464e-33+0.000000e+00i
 [8,]  5.637851e-18+0.000000e+00i -2.602085e-18+0.000000e+00i -7.155734e-18+6.938894e-18i -2.311116e-33-3.081488e-33i
 [9,]  2.602085e-18-1.734723e-18i  0.000000e+00-1.734723e-18i  1.734723e-18+0.000000e+00i -3.851860e-33+0.000000e+00i
[10,]  2.949030e-17+6.765422e-17i  8.673617e-19-3.469447e-18i  4.336809e-19-8.673617e-19i  1.540744e-33-9.244464e-33i
                            [,86]                       [,87]                       [,88]                       [,89]
 [1,] -1.734723e-18-2.602085e-18i  2.602085e-18+8.673617e-19i -1.734723e-18-1.561251e-17i  5.030698e-17+5.290907e-17i
 [2,]  1.431147e-17+1.474515e-17i -8.673617e-19+5.204170e-18i  0.000000e+00+1.040834e-17i  4.336809e-18+8.673617e-19i
 [3,] -8.673617e-19-1.734723e-18i -1.387779e-17-2.515349e-17i  1.734723e-18+3.469447e-18i  3.469447e-18-1.214306e-17i
 [4,] -1.344411e-17-1.734723e-18i  5.204170e-18-6.071532e-18i  2.168404e-17+2.688821e-17i  2.602085e-18+8.673617e-18i
 [5,]  3.035766e-18+0.000000e+00i -1.734723e-18-1.734723e-18i -5.204170e-18-1.734723e-18i  5.464379e-17+2.775558e-17i
 [6,] -2.818926e-17+2.602085e-17i  6.938894e-18-6.071532e-18i -8.673617e-19-1.040834e-17i -2.602085e-18-1.734723e-18i
 [7,] -1.301043e-18-6.071532e-18i -5.377643e-17+2.862294e-17i -8.673617e-19-1.734723e-18i  8.673617e-19-1.127570e-17i
 [8,] -2.602085e-18+2.602085e-18i  8.673617e-19-3.469447e-18i -7.112366e-17+5.724587e-17i  5.204170e-18+1.040834e-17i
 [9,]  1.084202e-18-1.734723e-18i  8.673617e-19+0.000000e+00i -8.673617e-19+3.469447e-18i  1.474515e-17-2.602085e-17i
[10,] -2.168404e-19+6.071532e-18i -4.336809e-19-3.469447e-18i  2.602085e-18-3.469447e-18i  6.071532e-18-1.734723e-18i
                            [,90]                       [,91]                       [,92]                       [,93]
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 [2,] -1.214306e-17+1.301043e-18i -8.628166e-32-3.081488e-32i  1.734723e-18-2.168404e-18i  2.081668e-17+1.734723e-18i
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 [7,]  2.602085e-18-2.602085e-18i  5.916457e-31+1.232595e-32i -8.673617e-19-1.734723e-18i  6.938894e-18+0.000000e+00i
 [8,] -2.602085e-18-6.938894e-18i  1.848893e-32-6.779273e-32i  3.989864e-17+2.515349e-17i -1.734723e-18-8.673617e-19i
 [9,]  3.035766e-18-8.673617e-19i -5.546678e-32+9.860761e-32i  4.336809e-18+0.000000e+00i  6.852158e-17+1.734723e-18i
[10,]  4.336809e-19-8.673617e-19i  0.000000e+00+6.162976e-33i  2.602085e-18+5.204170e-18i  1.734723e-18+7.806256e-18i
                            [,94]                       [,95]                       [,96]
 [1,] -1.040834e-17+2.168404e-18i  3.122502e-17+7.372575e-18i -1.561251e-17+5.009014e-17i
 [2,] -5.724587e-17-1.205633e-16i  8.673617e-18+4.163336e-17i -1.734723e-18-2.688821e-17i
 [3,] -2.081668e-17+2.602085e-18i  5.551115e-17-5.637851e-17i -1.734723e-18+1.214306e-17i
 [4,] -6.938894e-18+5.030698e-17i  0.000000e+00+1.040834e-17i  2.862294e-17-2.797242e-17i
 [5,] -6.938894e-18+3.469447e-18i  1.734723e-17+2.341877e-17i  8.673617e-19+5.637851e-18i
 [6,] -1.092876e-16+4.336809e-18i -1.734723e-18-1.040834e-17i  8.673617e-19-1.734723e-18i
 [7,]  3.469447e-18-1.561251e-17i -2.428613e-17+5.030698e-17i  3.469447e-18-3.903128e-18i
 [8,] -3.469447e-18+2.775558e-17i  1.127570e-17+8.673617e-18i -2.949030e-17+2.255141e-17i
 [9,]  0.000000e+00-1.734723e-18i -1.387779e-17-8.673617e-19i  2.602085e-18+4.770490e-18i
[10,]  2.775558e-17-3.209238e-17i -8.673617e-19-1.040834e-17i -8.673617e-19-1.301043e-17i
 [ достигнута getOption("max.print") -- пропущено 86 строк ]
plot(1:L, res_comp_wise[[1]][, 2])
Предупреждение в xy.coords(x, y, xlabel, ylabel, log) :
  мнимые части убраны при преобразовании

avr <- averaging(res_comp_wise)

for (i in 1:dim(res$t_series)[2]){
  plot(x, avr[i, ])
}
Ошибка в 1:dim(res$t_series)[2] :аргумент нулевой длины
# Ft %*% sweep(Ft_inv, 1, X[, 1], '*')[2, ]
reconstruct_fft <- function(x, y, frequencies) {
  # Выполняем быстрое преобразование Фурье
  fft_y <- fft(y)

  # Получаем амплитуды и фазы
  amplitudes <- Mod(fft_y)
  phases <- Arg(fft_y)
  
  reconstructed <- matrix(0, length(amplitudes), length(x))
  n <- length(amplitudes)
  for (i in 1:(length(amplitudes))) {
    # print(i)
    reconstructed[i, ] <-
      amplitudes[i] * 
      cos(2 * pi * frequencies[i] * (x) + phases[i]) /
      n
  }
  # plot(x, reconstructed[9, ], main = paste(x[1]), type = "l", col = "red")
  # lines(x, y/2)
  # plot(x, reconstructed[L+2-9, ], main = paste(x[1]))
  return(reconstructed)
}


n <- 96*2-1
x <- 0:(n-1)
L <- 48
frequencies <- (0:(L-1)) / L
y <- sin(2*pi/12 * x)
y_main <- y
x_main <- x

X <- hankel(y, L)
K <- dim(X)[2]
res <- list()
for (i in 1:K){
  y <- X[, i]
  x <- x_main[1:(1 + L - 1)]
  res[[i]] <- reconstruct_fft(x, y, frequencies)
}

# for (i in 1:L){
#   plot(x, res[[i]][9, ])
# }

res_mult <- res
# res

averaging <- function(res_comp_wise_mult){
  K <- dim(X)[2]
  counters <- rep(0, n)
  res <- matrix(0, ncol = n, nrow = L)
  for (i in 1:length(res_comp_wise_mult)){
    res[, i:(i+L-1)] <- res[, i:(i+L-1)] + res_comp_wise_mult[[i]]
    counters[i:(i+L-1)] <- counters[i:(i+L-1)] + 1
  }
  for (i in 1:n){
    res[, i] <- res[, i] / counters[i]
  }
  res
}

avr <- averaging(res_mult)

group_by_elementary_freq_foureir <- function(res_averaged){
  nf2 <- 0
  if (L %% 2) {
    nf2 <- (L + 1) / 2 - 1
  } else {
    nf2 <- L / 2 - 1
  }
  nft <- nf2 + abs((L %% 2) - 2)
  
  Z <- matrix(0, ncol = nft, nrow = n)
  
  # print(Z |> dim())
  # print(res_averaged |> dim())
  
  Z[, 1] <- res_averaged[1, ]
  for (k in 1:nf2) {
    Z[, k + 1] <- res_averaged[k + 1, ] + res_averaged[L + 2 - (k + 1), ]
  }
  if (L %% 2 != 0) {
    Z[, nft] <- res_averaged[nft, ]
  }
  
  
  return(list(
    t_series = Z,
    freq = (0:dim(Z)[2])/L
  ))
}


rs <- group_by_elementary_freq_foureir(avr)

plot(x_main, rs$t_series[, 5])

IP_values

data_slice <- 1:537
dates_slice <- dates[data_slice]
IP_values_slice <- IP_values[data_slice]
eps <- 1/193


c <- circulant_SSA(IP_values_slice, L = 192, extend_flag = FALSE)
r <- c$t_series






reconstruct_fft <- function(x, y, frequencies) {
  # Выполняем быстрое преобразование Фурье
  fft_y <- fft(y)

  # Получаем амплитуды и фазы
  amplitudes <- Mod(fft_y)
  phases <- Arg(fft_y)
  
  reconstructed <- matrix(0, length(amplitudes), length(x))
  n <- length(amplitudes)
  for (i in 1:(length(amplitudes))) {
    # print(i)
    reconstructed[i, ] <-
      amplitudes[i] * 
      cos(2 * pi * frequencies[i] * (x) + phases[i]) /
      n
  }
  # plot(x, reconstructed[9, ], main = paste(x[1]), type = "l", col = "red")
  # lines(x, y/2)
  # plot(x, reconstructed[L+2-9, ], main = paste(x[1]))
  return(reconstructed)
}


n <- 537
x <- 0:(n-1)
L <- 192
frequencies <- (0:(L-1)) / L
y <- IP_values_slice
y_main <- y
x_main <- x

X <- hankel(y, L)
K <- dim(X)[2]
res <- list()
for (i in 1:K){
  y <- X[, i]
  x <- x_main[1:(1 + L - 1)]
  res[[i]] <- reconstruct_fft(x, y, frequencies)
}

# for (i in 1:L){
#   plot(x, res[[i]][9, ])
# }

res_mult <- res
# res

averaging <- function(res_comp_wise_mult){
  K <- dim(X)[2]
  counters <- rep(0, n)
  res <- matrix(0, ncol = n, nrow = L)
  for (i in 1:length(res_comp_wise_mult)){
    res[, i:(i+L-1)] <- res[, i:(i+L-1)] + res_comp_wise_mult[[i]]
    counters[i:(i+L-1)] <- counters[i:(i+L-1)] + 1
  }
  for (i in 1:n){
    res[, i] <- res[, i] / counters[i]
  }
  res
}

avr <- averaging(res_mult)

group_by_elementary_freq_foureir <- function(res_averaged){
  nf2 <- 0
  if (L %% 2) {
    nf2 <- (L + 1) / 2 - 1
  } else {
    nf2 <- L / 2 - 1
  }
  nft <- nf2 + abs((L %% 2) - 2)
  
  Z <- matrix(0, ncol = nft, nrow = n)
  
  # print(Z |> dim())
  # print(res_averaged |> dim())
  
  Z[, 1] <- res_averaged[1, ]
  for (k in 1:nf2) {
    Z[, k + 1] <- res_averaged[k + 1, ] + res_averaged[L + 2 - (k + 1), ]
  }
  if (L %% 2 != 0) {
    Z[, nft] <- res_averaged[nft, ]
  }
  
  
  return(list(
    t_series = Z,
    freq = (0:dim(Z)[2])/L
  ))
}


rs <- group_by_elementary_freq_foureir(avr)

# plot(x_main, rs$t_series[, 5])


for (i in 1:dim(r)[2]){
  plot(1:n, rs$t_series[, i], col= "red", type = "l")
  lines(1:n, r[, i])
}
cissa_like_fourier_transform <- function(ts, L){
  reconstruct_fft <- function(x, y, frequencies) {
    # Выполняем быстрое преобразование Фурье
    fft_y <- fft(y)
  
    # Получаем амплитуды и фазы
    amplitudes <- Mod(fft_y)
    phases <- Arg(fft_y)
    
    reconstructed <- matrix(0, length(amplitudes), length(x))
    n <- length(amplitudes)
    for (i in 1:(length(amplitudes))) {
      # print(i)
      reconstructed[i, ] <-
        amplitudes[i] * 
        cos(2 * pi * frequencies[i] * (x) + phases[i]) /
        n
    }
    # plot(x, reconstructed[9, ], main = paste(x[1]), type = "l", col = "red")
    # lines(x, y/2)
    # plot(x, reconstructed[L+2-9, ], main = paste(x[1]))
    return(reconstructed)
  }
  
  
  n <- length(ts)
  x <- 0:(n-1)
  L <- L
  frequencies <- (0:(L-1)) / L
  y <- ts
  y_main <- y
  x_main <- x
  
  X <- hankel(y, L)
  K <- dim(X)[2]
  res <- list()
  for (i in 1:K){
    y <- X[, i]
    x <- x_main[1:(1 + L - 1)]
    res[[i]] <- reconstruct_fft(x, y, frequencies)
  }
  
  # for (i in 1:L){
  #   plot(x, res[[i]][9, ])
  # }
  
  res_mult <- res
  # res
  
  averaging <- function(res_comp_wise_mult){
    K <- dim(X)[2]
    counters <- rep(0, n)
    res <- matrix(0, ncol = n, nrow = L)
    for (i in 1:length(res_comp_wise_mult)){
      res[, i:(i+L-1)] <- res[, i:(i+L-1)] + res_comp_wise_mult[[i]]
      counters[i:(i+L-1)] <- counters[i:(i+L-1)] + 1
    }
    for (i in 1:n){
      res[, i] <- res[, i] / counters[i]
    }
    res
  }
  
  avr <- averaging(res_mult)
  
  group_by_elementary_freq_foureir <- function(res_averaged){
    nf2 <- 0
    if (L %% 2) {
      nf2 <- (L + 1) / 2 - 1
    } else {
      nf2 <- L / 2 - 1
    }
    nft <- nf2 + abs((L %% 2) - 2)
    
    Z <- matrix(0, ncol = nft, nrow = n)
    
    # print(Z |> dim())
    # print(res_averaged |> dim())
    
    Z[, 1] <- res_averaged[1, ]
    for (k in 1:nf2) {
      Z[, k + 1] <- res_averaged[k + 1, ] + res_averaged[L + 2 - (k + 1), ]
    }
    if (L %% 2 != 0) {
      Z[, nft] <- res_averaged[nft, ]
    }
    
    
    return(list(
      t_series = Z,
      freq = (0:dim(Z)[2])/L
    ))
  }


  rs <- group_by_elementary_freq_foureir(avr)
  return(rs)
}
data_slice <- 1:537
dates_slice <- dates[data_slice]
IP_values_slice <- IP_values[data_slice]
eps <- 1/193


c <- circulant_SSA(IP_values_slice, L = 192, extend_flag = FALSE)
r <- c$t_series

c_ft <- cissa_like_fourier_transform(IP_values_slice, L = 192)
r_ft <- c_ft$t_series


for (i in 1:dim(r)[2]){
  plot(1:n, r_ft[, i], col= "red", type = "l", lwd = 2)
  lines(1:n, r[, i])
}

---
title: "Circulant SSA"
author: "Погребников Николай"
output: html_notebook
---

## Вспомогательные функции

```{r}
library(Rssa)
library(signal)
library(gsignal)
source("eossa_new.R")


dftmtx <- function(n) {
  y <- stats::mvfft(diag(1, n))
  y
}

diag_averaging <- function(A){
  B <- A[nrow(A):1, ] |> Re()
  lapply(split(B, -(row(B) - col(B)) ), mean) |> as.numeric()
}

shift_vector <- function(vec) {
  last_element <- tail(vec, 1)
  vec <- vec[-length(vec)]
  shifted_vec <- c(last_element, vec)
  return(shifted_vec)
}

extend <- function(x, H){
  # Вычисление коэффициентов AR модели для дифференцированного ряда
  N <- length(x)
  p <- floor(N / 3)
  dx <- diff(x)
  # A <- ar(dx, order.max = p, method = "yule-walker")$ar
  A <- aryule(dx, p)$a
  
  # Правое расширение
  y <- x
  dy <- diff(y)
  er <- signal::filter(A, 1, dy)
  dy <- signal::filter(1, A, c(er, rep(0, H)))
  y <- y[1] + c(0, cumsum(dy))
  
  # Левое расширение
  y <- rev(y)
  dy <- diff(y)
  er <- signal::filter(A,1,dy)
  dy <- signal::filter(1,A,c(er, rep(0, H)))
  y <- y[1] + c(0, cumsum(dy))
  
  # Расширенный ряд
  xe <- rev(y)
  
  # Вывод результатов
  xe 
}
```

## CiSSA

Подаётся на вход временной ряд, длина окна (если её нет, то она равна длине ряда + 1 пополам) и информация о том, нужно ли расширить ряд. Расширять ряд стоит при стохастическом тренде (Autoregressive extension (default). It is indicated for stationary and stochastic trend time series as well). Реализовано только Autoregressive extension.

\
На выходе список выдаётся список list(t_series, importance).\
t_series — матрица, по столбцам которой располагаются временные ряды, отвечающие за частоты (i-1)/L, где i — номер столбца, L — длина окна.\
importance — вектор, отвечающий за значимость i-ого временного ряда в разлолжении. Чем больше значение, тем больший вклад внёс i-тый временной ряд.

```{r}
circulant_SSA <- function(ts, L = NULL, extend_flag = FALSE){
  time_series <- ts
  # Construct trajectory matrix
  N <- length(time_series)
  if (is.null(L)){
    L <- (N + 1)%/%2
  }
  # Проверка на расширения ряда
  if (extend_flag == FALSE){
    H <- 0
    time_series <- ts
  }
  else{
    H <- L
    time_series <- extend(ts, H)
  }
  
  X <- hankel(time_series, L)
  
  # Number of symmetric frequency pairs around 1/2
  if (L %% 2) {
    nf2 <- (L + 1) / 2 - 1
  } else {
    nf2 <- L / 2 - 1
  }
  
  # Number of frequencies <= 1/2
  nft <- nf2 + abs((L %% 2) - 2)
  
  # Decomposition
  # Estimate autocovariance     OK
  autocov <- numeric(L)
  for (m in 0:(L-1)){
    autocov[[m+1]] <- sum(time_series[1:(N-m)] * time_series[(1+m):N]) / (N-m)
  }
  
  # First row of circulant matrix
  circ_first_row <- numeric(L)
  for (m in 0:(L-1)){
    circ_first_row[[m+1]] <- (L-m)/L * autocov[[m+1]] + (m)/L * autocov[[L-m]]
  }
  
  # Build circulant matrix
  S_C <- matrix(circ_first_row, nrow = 1)
  shifted_vector <- circ_first_row
  for (i in 2:(L)) {
    shifted_vector <- shift_vector(shifted_vector)
    # S_C <- rbind(S_C, as.vector(shifted_vector))
    S_C <- rbind(as.vector(shifted_vector), S_C)
  }
  
  # Eigenvectors of circulant matrix (unitary base)
  U <- dftmtx(L)/sqrt(L)
  
  # Real eigenvectors (orthonormal base)
  U[, 1] <- Re(U[, 1])
  for (k in 1:nf2) {
    u_k <- U[, k + 1]
    U[, k + 1] <- sqrt(2) * Re(u_k)
    U[, L + 2 - (k + 1)] <- sqrt(2) * Im(u_k)
  }
  if (L %% 2 != 0) {
    U[, nft] <- Re(U[, nft])
  }
  
  # Eigenvalues of circulant matrix: estimated power spectral density
  psd <- abs(diag(t(U) %*% S_C %*% U))
  
  # Principal components
  W <- t(U) %*% X
  # Reconstruction
  # Elementary reconstructed series
  R <- matrix(0, nrow = N+2*H, ncol = L)
  for (k in 1:L) {
    R[, k] <- U[ ,k] %*% t(W[k, ]) |> diag_averaging()
  }
  
  # Grouping by frequency
  # Elementary reconstructed series by frequency
  Z <- matrix(0, nrow = N+2*H, ncol = nft)
  Z[, 1] <- R[, 1]
  # Importance of component
  imp <- numeric(nft)
  lambda_sm <- sum(psd)
  imp[1] <- psd[1]/lambda_sm
  for (k in 1:nf2) {
    Z[, k + 1] <- R[, k + 1] + R[, L + 2 - (k + 1)]
    imp[k+1] <- (psd[k+1] + psd[ L + 2 - (k + 1)])/lambda_sm
  }
  if (L %% 2 != 0) {
    Z[, nft] <- R[, nft]
    imp[nft] <- psd[nft] / lambda_sm
  }
  
  list(t_series = Z[(H+1):(N+H),],
       importance = imp,
       freq = (0:(length(imp) -1))/L
       )
}
```

```{r}
# groups - list of frequencies
grouping_cissa <- function(cissa_res, groups){
  freq <- cissa_res$freq
  t_series <- cissa_res$t_series
  
  residuals <- 0
  result <- setNames(as.list(rep(0, length(groups))), names(groups))
  for (i in 1:length(cissa_res$freq)){
    flag <- FALSE
    for (name in names(groups)){
      if (groups[[name]][1] <= freq[i] & freq[i] <= groups[[name]][2]){
        flag <- TRUE
        result[[name]] <- result[[name]] + t_series[, i]
      }
    }
    
    if (flag == FALSE){
      residuals <- residuals + t_series[, i]
    }
  }
  
  result[["residuals"]] <- residuals
  result
}
```

```{r}
generate_ts <- function(func, n=1e3, ...){
  1:n |> func(...) |> ts()
}

f_cos <- function(x, A = 1, omega = 1/4, phi = 0){
  f_exp_mod_harm_series(x, A, alpha = 0, omega = omega, phi = phi)
}

f_sin <- function(x, A = 1, omega = 1/4, phi = 3*pi/2){
  f_exp_mod_harm_series(x, A, alpha = 0, omega = omega, phi = phi)
}

f_exp <- function(x, A = 1, alpha = 1){
  A * exp(alpha * x)
}

f_exp_cos <- function(x, A = 1, alpha = 1, omega = 1/4, phi = 0){
  f_exp_mod_harm_series(x, A, alpha, omega, phi)
}

f_const <- function(x, C = 0){
  rep(C, length(x))
}

f_exp_mod_harm_series <- function(x, A = 1, alpha = 1, omega = 1/4, phi = 0){
  A*exp(alpha*x)*cos(2*pi*omega*x + phi)
}

f_linear <- function(x, a = 1, b = 0){
  a*x + b
}
mse <- function(f_true, f_reconstructed){
   mean((f_true - f_reconstructed)^2) 
}
```

#### Ошибка при Lw in N, Kw not in N

```{r}
n <- 96*2+5
L <- 96
f_sum <- function(x){
  f_const(x, C = 1) + f_cos(x, omega = 1/12) 
}


f_const |> generate_ts(n, C = 1) |>
  plot(col = "green", ylim = c(-1, 2), ylab = "f_n")
f_cos |>
  generate_ts(n, omega = 1/12) |>
  lines(col="green")
f_sum |> generate_ts(n) |> lines(lwd = 3, col='red')
f_n <- f_sum(1:n)



c <- circulant_SSA(f_n, L = 96, extend_flag = FALSE)
r <- grouping_cissa(c,
               groups = list(
                 trend = c(0, 1/100),
                 sesonal = c(1/99, 1/10)
               )
               )

f_C <- f_const |> generate_ts(n, C = 1)
f_c <- f_cos |> generate_ts(n, omega = 1/12)
print("Ошибки при CiSSA")
print(paste("Ошибка при вычислении C = 1: ", mse(f_C, r$trend) |> format(scientific = TRUE, digits = 2) ))
print(paste("Ошибка при вычислении cos(pi/12): ", mse(f_c, r$sesonal) |> format(scientific = TRUE, digits = 2) ))

lines(1:n, r$trend, col="blue")
lines(1:n, r$sesonal, col="blue")

f_const |> generate_ts(n, C = 1) |>
  plot(col = "green", ylim = c(-1, 2), ylab = "f_n")
f_cos |>
  generate_ts(n, omega = 1/12) |>
  lines(col="green")
f_sum |> generate_ts(n) |> lines(lwd = 3, col='red')
f_n <- f_sum(1:n)

s <- ssa(f_n, L = 96)
r <- reconstruct(s, groups=list(
  trend = 1,
  sesonal = 2:3
))


print("Ошибки при SSA")
print(paste("Ошибка при вычислении C = 1: ", mse(f_C, r$trend) |> format(scientific = TRUE, digits = 2)  ))
print(paste("Ошибка при вычислении cos(pi/12): ", mse(f_c, r$sesonal) |> format(scientific = TRUE, digits = 2)))

lines(1:n, r$trend)
lines(1:n, r$sesonal)
```

#### Проверка разделимости непериодических компонент + автогруппировка SSA

```{r}
n <- 96*2-1
L <- 96

C <- 1
omega_cs <- 1/12
omega_sn <- 1/24
a <- 1/100
f_sum <- function(x){
  f_const(x, C = C) +
    f_cos(x, omega = omega_cs) +
    f_exp(x, a = a) +
    f_sin(x, omega = omega_sn)
}


f_C <- f_const |> generate_ts(n, C = C)
f_c <- f_cos |> generate_ts(n, omega = omega_cs)
f_s <- f_sin |> generate_ts(n, omega = omega_sn)
f_e <- f_exp |> generate_ts(n, a = a)

f_n <- f_sum(1:n)

library(xtable)

# Шаг 2: Создание примера данных
data <- data.frame(
  Метод = c("SSA", "CiSSA"),
  e_err = c(20, 20),
  c_err = c(23, 35),
  ec_err = c(20, 20),
  sin_err = c (20, 20),
  cos_err = c(1, 1)
)


# Отрисовка ряда f_n
plot(f_n, type = "l", lwd = 3, col = 'red', ylim = c(-2, 10),
     xlab = "Время", ylab = "Значения ряда", main = "Разложение временного ряда")

# Добавление отдельных компонентов (f_C, f_c, f_e)
lines(f_C, col = "blue")  # Компонент f_C
lines(f_c, col = "blue")  # Компонент f_c
lines(f_e, col = "blue")  # Компонент f_e
lines(f_s, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)








c <- circulant_SSA(f_n, L = L, extend_flag = TRUE)
# r <- c$t_series
r <- grouping_cissa(c,
                    groups = list(
                      # trend = c(0, 1/100),
                      trend = c(0, 1/1000),
                      sesonal_cos = c(1/14, 1/10),
                      sesonal_sin = c(1/26, 1/23)
                    ))

data$cos_err[2] <- mse(f_s, r$sesonal_sin) |> formatC(format = "e", digits = 1)
data$sin_err[2] <- mse(f_c, r$sesonal_cos) |> formatC(format = "e", digits = 1)
data$ec_err[2] <- mse(f_C+f_e, r$trend) |> formatC(format = "e", digits = 1)


# png("C:/Users/nik1m/Desktop/уник/6 сем/курсач/Текст работы/img/trend inseparability/CiSSA.png")  # сохранение в формате PNG

plot(1:n, f_n, type = "l", lwd=3, ylim= c(-2, 10), col="red",
     xlab = "Время", ylab = "Значения ряда", main = "CiSSA разложение временного ряда")
lines(1:n, r$trend, col = "blue")
lines(1:n, r$sesonal_sin, col = "blue")
lines(1:n, r$sesonal_cos, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)

# dev.off()  # завершение сохранения










s <- ssa(f_n, L)
e <- eossa(s, 1:10, k = 7)

g_sesonal <- grouping.auto(e, base = "eigen",
                   freq.bins = list(trend = c(0.001),
                                    sesonal2 = c(1/25, 1/23),
                                    sesonal1 = c(1/13, 1/11)
                                    ),
                   threshold = 0.1)


r <- Rssa::reconstruct(e, groups=c(list(exp = 1,
                                C = 2
                                ),
                             g_sesonal)
                 )

plot(wcor(e, groups = 1:24), scales = list(at = c(10, 20, 30)))

data$c_err[1] <- mse(f_C, r$C) |> formatC(format = "e", digits = 1)
data$e_err[1] <- mse(f_e, r$exp) |> formatC(format = "e", digits = 1)
data$cos_err[1] <- mse(f_c, r$sesonal1) |> formatC(format = "e", digits = 1)
data$sin_err[1] <- mse(f_s, r$sesonal2) |> formatC(format = "e", digits = 1)
data$ec_err[1] <- mse(f_C+f_e, r$C+r$exp) |> formatC(format = "e", digits = 1)


# png("C:/Users/nik1m/Desktop/уник/6 сем/курсач/Текст работы/img/trend inseparability/SSA.png")  # сохранение в формате PNG

plot(1:n, f_n, type = "l", lwd=3, ylim= c(-2, 10), col="red",
     xlab = "Время", ylab = "Значения ряда", main = "SSA разложение временного ряда")

lines(1:n, r$trend, type = "l", col="green")
lines(1:n, r$exp, type = "l", ylim= c(-2, 10), col="blue")
lines(1:n, r$C, col = "blue")
lines(1:n, r$sesonal1, col = "blue")
lines(1:n, r$sesonal2, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)







# Шаг 3: Преобразование данных в формат LaTeX
table_latex <- xtable(data, caption = "Example Table")

# Шаг 4: Вывод таблицы в LaTeX файл
print(table_latex, include.rownames = FALSE)





```

#### Пример cos\*exp

```{r}
n <- 96*2-1
L <- 96
eps <- 1/(L+1)

C <- 1
omega_cs <- 1/12
omega_sn <- 1/24
a <- 1/100
omega_exp <- 1/48
f_sum <- function(x){
    f_cos(x, omega = omega_cs) +
    f_exp_mod_harm_series(x, a = a, omega = omega_exp) +
    f_sin(x, omega = omega_sn)
}


f_c <- f_cos |> generate_ts(n, omega = omega_cs)
f_s <- f_sin |> generate_ts(n, omega = omega_sn)
f_e <- f_exp_mod_harm_series |> generate_ts(n, a = a, omega = omega_exp)

f_n <- f_sum(1:n)

library(xtable)

# Шаг 2: Создание примера данных
data <- data.frame(
  Метод = c("SSA", "CiSSA"),
  exp_err = c(20, 20),
  sin_err = c (20, 20),
  cos_err = c(1, 1)
)


# Отрисовка ряда f_n
plot(f_n, type = "l", lwd = 3, col = 'red', ylim = c(-10, 10),
     xlab = "Время", ylab = "Значения ряда", main = "Разложение временного ряда")

# Добавление отдельных компонентов (f_C, f_c, f_e)
lines(f_c, col = "blue")  # Компонент f_c
lines(f_e, col = "blue")  # Компонент f_e
lines(f_s, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)








c <- circulant_SSA(f_n, L = L, extend_flag = TRUE)
# r <- c$t_series
r <- grouping_cissa(c,
                    groups = list(
                      trend = c(0, 1/26-eps),
                      sesonal_cos = c(1/14, 1/10),
                      sesonal_sin = c(1/26, 1/23)
                    ))

data$cos_err[2] <- mse(f_s, r$sesonal_sin) |> formatC(format = "e", digits = 1)
data$sin_err[2] <- mse(f_c, r$sesonal_cos) |> formatC(format = "e", digits = 1)
data$exp_err[2] <- mse(f_e, r$trend) |> formatC(format = "e", digits = 1)


# png("C:/Users/nik1m/Desktop/уник/6 сем/курсач/Текст работы/img/trend inseparability/CiSSA.png")  # сохранение в формате PNG

plot(1:n, f_n, type = "l", lwd=3, ylim= c(-10, 10), col="red",
     xlab = "Время", ylab = "Значения ряда", main = "CiSSA разложение временного ряда")
lines(1:n, r$trend, col = "blue")
lines(1:n, r$sesonal_sin, col = "blue")
lines(1:n, r$sesonal_cos, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)

# dev.off()  # завершение сохранения










s <- ssa(f_n, L)
e <- eossa_new(s, nested.groups = list(1:30), clust_type = "distance")

g_sesonal <- grouping.auto(e, base = "eigen",
                   freq.bins = list(trend = c(1/25-eps),
                                    sesonal2 = c(1/25, 1/23),
                                    sesonal1 = c(1/13, 1/11)
                                    ),
                   threshold = 0.1)


r <- reconstruct(e, groups= g_sesonal)

plot(wcor(e, groups = 1:24), scales = list(at = c(10, 20, 30)))

data$exp_err[1] <- mse(f_e, r$trend)  |> formatC(format = "e", digits = 1)
data$cos_err[1] <- mse(f_c, r$sesonal1) |> formatC(format = "e", digits = 1)
data$sin_err[1] <- mse(f_s, r$sesonal2) |> formatC(format = "e", digits = 1)


# png("C:/Users/nik1m/Desktop/уник/6 сем/курсач/Текст работы/img/trend inseparability/SSA.png")  # сохранение в формате PNG

plot(1:n, f_n, type = "l", lwd=3, ylim= c(-10, 10), col="red",
     xlab = "Время", ylab = "Значения ряда", main = "SSA разложение временного ряда")

lines(1:n, r$trend, type = "l", col="blue")
lines(1:n, r$sesonal1, col = "blue")
lines(1:n, r$sesonal2, col = "blue")

# Легенда
legend("topleft", legend = c("Весь ряд", "Компоненты"), 
       col = c("red", "blue"), lty = 1, lwd = 3)







# Шаг 3: Преобразование данных в формат LaTeX
table_latex <- xtable(data, caption = "Example Table")

# Шаг 4: Вывод таблицы в LaTeX файл
print(table_latex, include.rownames = FALSE)


```

## Данные IP

```{r}
library(readxl)
data <- read_excel("Data/International_Financial_Statistics_.xlsx")
data |> head()
```

Отрисовка данных IP

```{r}
dates <- seq(as.Date("1970-01-01"), as.Date("2018-1-30"), by = "month")
IP_values <- data[2, -c(1, 2)] |> as.double() 
plot(dates, IP_values, type="l")
```

#### Cissa

Отрисовка трендовой составляющей чёрным цветом, основной временной ряд — красным

```{r}
data_slice <- 1:537
dates_slice <- dates[data_slice]
IP_values_slice <- IP_values[data_slice]
eps <- 1/193

c <- circulant_SSA(IP_values_slice, L = 192, extend_flag = TRUE)
r <- c$t_series
r <- grouping_cissa(c,
                    groups = list(
                      trend = c(0, 1/192),
                      cycle = c(1/97, 5/95),
                      sesonal = c(1/13, 1/2+0.0001)
                    )
                    )
r_sesonal <-  grouping_cissa(c,
                             groups = list(
                              s1 = c(16/192 - eps, 16/192 + eps),
                              s2 = c(32/192 - eps, 32/192 + eps),
                              s3 = c(48/192 - eps, 48/192 + eps),
                              s4 = c(64/192 - eps, 64/192 + eps),
                              s5 = c(80/192 - eps, 80/192 + eps),
                              s6 = c(96/192 - eps, 96/192 + eps)
                             )
                             )
# cissa_trend <- r[,1] + r[,2]
# cissa_cycle <- r[, 3:11] |> rowSums()
# cissa_sesonal <- r[, c(17, 33, 49, 65, 81, 97)] |> rowSums()
# cissa_residuals <- IP_values_slice - (cissa_trend + cissa_cycle + cissa_sesonal)

cissa_trend <- r$trend
cissa_cycle <- r$cycle
cissa_sesonal <- Reduce("+", r_sesonal |> within(rm(residuals)))
cissa_residuals <- IP_values_slice - (cissa_trend + cissa_cycle + cissa_sesonal)


plot(dates_slice, IP_values_slice,
     type="l", col = "black")
lines(dates_slice, cissa_trend,
      type="l", col = "red")

plot(dates_slice, cissa_cycle,
     type="l", col = "red")

plot(dates_slice, cissa_sesonal,
     type="l", col = "red")

plot(dates_slice, cissa_residuals,
     type="l", col = "red")

plot(dates_slice, IP_values_slice,
     type="l", col = "black")
lines(dates_slice, cissa_trend+cissa_cycle+cissa_sesonal,
      type="l", col = "red")
```

#### SSA fossa

```{r}
s <- ssa(IP_values_slice, L = 192)
e <- fossa(s)
# e <- eossa_new(s, nested.groups = list(1:30), clust_type = "distance")
eps <- 1/193

groups <- grouping.auto(e,
                   freq.bins = list(trend = c(1/192),
                                    cycle = c(1/97, 5/95),
                                    s1 = c(16/192 - eps, 16/192 + eps),
                                    s2 = c(32/192 - eps, 32/192 + eps),
                                    s3 = c(48/192 - eps, 48/192 + eps),
                                    s4 = c(64/192 - eps, 64/192 + eps),
                                    s5 = c(80/192 - eps, 80/192 + eps),
                                    s6 = c(96/192 - eps, 96/192 + eps)
                                    ),
                   threshold = 0)


plot(wcor(e, groups = 1:30), scales = list(at = c(10, 20, 30)),
     main = "W-correlation matrix SSA (fossa)")

r <- reconstruct(e, groups=groups)

ssa_trend_f <- r$trend
ssa_cycle_f <- r$cycle
ssa_sesonal_f <- r$s1 + r$s2 + r$s3 + r$s4 + r$s5 + r$s6
ssa_residuals_f <- IP_values_slice - (ssa_trend_f + ssa_cycle_f + ssa_sesonal_f)

plot(dates_slice, IP_values_slice,
     type="l", col = "black")
lines(dates_slice, ssa_trend_f,
      type="l", col = "magenta")

plot(dates_slice, ssa_cycle_f, 
     type="l", col = "magenta")

plot(dates_slice, ssa_sesonal_f, 
     type="l", col = "magenta")

plot(dates_slice, ssa_residuals_f,
     type="l", col = "magenta")

```

#### SSA eossa

```{r}
library(Rssa)
source("eossa_new.r")
s <- ssa(IP_values_slice, L = 192)
e <- eossa_new(s, nested.groups = list(1:30), clust_type = "distance")




groups <- grouping.auto(e,
                   freq.bins = list(trend = c(1/192),
                                    cycle = c(1/97, 5/95),
                                    s1 = c(16/192 - eps, 16/192 + eps),
                                    s2 = c(32/192 - eps, 32/192 + eps),
                                    s3 = c(48/192 - eps, 48/192 + eps),
                                    s4 = c(64/192 - eps, 64/192 + eps),
                                    s5 = c(80/192 - eps, 80/192 + eps),
                                    s6 = c(96/192 - eps, 96/192 + eps)
                                    ),
                   threshold = 0)
plot(wcor(e, groups = 1:30), scales = list(at = c(10, 20, 30)),
     main = "W-correlation matrix SSA (eossa)")

r <- reconstruct(e, groups=groups)

ssa_trend <- r$trend
ssa_cycle <- r$cycle
ssa_sesonal <- r$s1 + r$s2 + r$s3 + r$s4 + r$s5 + r$s6
ssa_residuals <- IP_values_slice - (ssa_trend + ssa_cycle + ssa_sesonal)

plot(dates_slice, IP_values_slice,
     type="l", col = "black")
lines(dates_slice, ssa_trend,
      type="l", col = "blue")

plot(dates_slice, ssa_cycle, 
     type="l", col = "blue")

plot(dates_slice, ssa_sesonal, 
     type="l", col = "blue")

plot(dates_slice, ssa_residuals,
     type="l", col = "blue")
```

```{r}
plot(dates_slice, IP_values_slice,
     main = "IP USA тренд",xlab = "Время", ylab = "Значение",
     type="l", col = "black")
lines(dates_slice, ssa_trend,
      type="l", col = "blue", lwd=2)
lines(dates_slice, ssa_trend_f,
      type="l", col = "magenta", lwd=2)
lines(dates_slice, cissa_trend,
      type="l", col = "red", lwd=2)
# Легенда
legend("topleft", legend = c("Весь ряд", "CiSSA тренд", "SSA тренд (eossa)", "SSA тренд (fossa)"), 
       col = c("black", "red", "blue", "magenta"), lty = 1, lwd = 3)


plot(dates_slice, ssa_cycle,
     main = "IP USA цикличность", xlab = "Время", ylab = "Значение",
     type="l", col = "blue", ylim=c(-10, 10), lwd=2)
lines(dates_slice, cissa_cycle,
      type="l", col = "red", lwd=2)
lines(dates_slice, ssa_cycle_f,
      type="l", col = "magenta", lwd=2)
# Легенда
legend("topleft", legend = c("CiSSA", "SSA (eossa)", "SSA (fossa)"), 
       col = c("red", "blue", "magenta"), lty = 1, lwd = 3)

```

```{r}
# Настройка графиков для отображения двух графиков один под другим с общей осью X
layout(matrix(c(1, 2), nrow = 2, byrow = TRUE), heights = c(1, 1.2))

# Построение первого графика
par(mar = c(2, 4, 2, 2)) # Уменьшение нижнего отступа
plot(dates_slice, ssa_sesonal, type = "l", col = "blue", lwd = 1,
     main = "SSA (eossa) сезонность", xlab = "", ylab = "Значение")
# Добавление оси X внизу первого графика, но с пустыми метками
axis(1, labels = FALSE)

# Построение второго графика
par(mar = c(5, 4, 2, 2)) # Увеличение нижнего отступа
plot(dates_slice, ssa_sesonal_f, type = "l", col = "magenta", lwd = 1,
     main = "SSA (fossa) сезонность", xlab = "Время", ylab = "Значение")

par(mar = c(3, 4, 2, 2)) # Увеличение нижнего отступа
plot(dates_slice, cissa_sesonal, type = "l", col = "red", lwd = 1,
     main = "CiSSA сезонность", xlab = "Время", ylab = "Значение")

# Восстановление макета по умолчанию
layout(1)


```

```{r}
plot(dates_slice, ssa_residuals, 
     main = "IP USA остаток", xlab = "Время", ylab = "Значение",
     type="l", col = "blue", ylim=c(-2, 2))
lines(dates_slice, cissa_residuals,
      type="l", col = "red")
lines(dates_slice, ssa_residuals_f,
      type="l", col = "magenta")
legend("topleft", legend = c("CiSSA", "SSA (eossa)", "SSA (fossa)"), 
       col = c("red", "blue", "magenta"), lty = 1, lwd = 3)
```

```{r}
ssa_residuals |> density() |> plot()
cissa_residuals |> density() |> plot()
```

### Отделение сигнала от шума

```{r}
set.seed(100)

n_mse_tests <- function(n){
  n <- 96*2-1
  L <- 96
  sigma <- 0.1
  
  
  C <- 1
  omega_cs <- 1/12
  omega_sn <- 1/24
  a <- 1/100
  f_sum <- function(x){
    f_const(x, C = C) +
      f_cos(x, omega = omega_cs) +
      f_exp(x, a = a) +
      f_sin(x, omega = omega_sn)
  }
  
  
  f_C <- f_const |> generate_ts(n, C = C)
  f_c <- f_cos |> generate_ts(n, omega = omega_cs)
  f_s <- f_sin |> generate_ts(n, omega = omega_sn)
  f_e <- f_exp |> generate_ts(n, a = a)
  
  mse_lst <- list()
  for (i in 1:n) {
    f_noise <- rnorm(n, sd = sigma)
    
    f_n <- f_sum(1:n) + f_noise
    
    
    
    c <- circulant_SSA(f_n, L = L, extend_flag = TRUE)
    # r <- c$t_series
    r <- grouping_cissa(c, groups= list(trend = c(0, 1/1000), 
                                        sesonal2 = c(1/25, 1/23),
                                        sesonal1 = c(1/13, 1/10)
                                        ))
    
    # mse_lst$cissa <- c(mse_lst$cissa, mse(f_sum(1:n), r[, 9] + r[, 5] + r[, 1])) 
    mse_lst$cissa <- c(mse_lst$cissa,
                       mse(f_sum(1:n),
                           r$trend + r$sesonal1 + r$sesonal2)) 
    
    
    
    
    s <- ssa(f_n, L)
    # e <- eossa(s, 1:10, k = 6)
    e <- fossa(s)
    
    g_sesonal <- grouping.auto(e, base = "eigen",
                       freq.bins = list(trend = 1/1000, 
                                        sesonal2 = c(1/25, 1/23),
                                        sesonal1 = c(1/13, 1/10)
                                        ),
                       threshold = 0.5)
    
    r <- reconstruct(e, groups=c(list(exp = 1, C = 2), g_sesonal))
    
    mse_lst$ssa <- 
      c(mse_lst$ssa, mse(f_sum(1:n), r$trend + r$sesonal2 + r$sesonal1))
 
  }
  return(mse_lst)
}

res_mse_test <- n_mse_tests(10000)

```

```{r}
# Оценка плотности
density_estimate_cissa <- density(res_mse_test$cissa)

# Построение графика плотности
plot(density_estimate_cissa, main = "Оценка плотности", 
     xlab = "Значение", ylab = "Плотность", 
     col = "blue", lwd = 2)

density_estimate_ssa <- density(res_mse_test$ssa)

# Построение графика плотности
plot(density_estimate_ssa, main = "Оценка плотности", 
     xlab = "Значение", ylab = "Плотность", 
     col = "blue", lwd = 2)

res_mse_test$cissa |> summary()
res_mse_test$cissa |> sd()
res_mse_test$ssa |> summary()
res_mse_test$ssa |> sd()
```

### Как выполняется расширение ряда

```{r}
IP_values_slice |> extend(192) |> plot(type="l", lwd = 3)
c(rep(0, 192),IP_values_slice) |> lines(type="l", col="red")

```

### Фурье преобразование

```{r}
n <- 96*2
L <- 96

x <- 0:(n-1)
y1 <- cos(2 * pi / 12 * x)  # Первая компонента
y2 <- sin(2 * pi / 48 * x)  # Вторая компонента

# Создаем общий временной ряд
y <- y1 + y2

# Выполняем быстрое преобразование Фурье
fft_y <- fft(y)

# Получаем амплитуды и фазы
amplitudes <- Mod(fft_y)
phases <- Arg(fft_y)

# Индексы для частот
n <- length(y)
frequencies <- (0:(n-1)) / n

# Функция для восстановления компонент
reconstruct_fft <- function(frequencies, amplitudes, phases) {
  reconstructed <- list()
  n <- length(amplitudes)
  for (i in 1:(length(amplitudes))) {
    reconstructed[[i]] <-
      amplitudes[i] * cos(2 * pi * frequencies[i] * x + phases[i]) / n * 2
  }
  return(reconstructed)
}

# y_main <- y
# X <- hankel(y)
# K <- dim(X)[2]
# res <- list()
# for (i in 1:K){
#   y <- X[, i]
#   
#   # Выполняем быстрое преобразование Фурье
#   fft_y <- fft(y)
#   
#   # Получаем амплитуды и фазы
#   amplitudes <- Mod(fft_y)
#   phases <- Arg(fft_y)
#   
#   res[[i]] <- reconstruct_fft(frequencies, amplitudes, phases)
# }
# 
# nft <- res[[1]]
# print(nft |> length())
# res_full <- matrix(0, nrow =  |> length(), )
# 
# for (i in 1:K){
#   res_full <- 
# }

# 
# 
y_reconstructed <- reconstruct_fft(frequencies, amplitudes, phases)

# Строим графики
for (i in 1:(n)){
  plot(x, y_reconstructed[[i]], main = paste(frequencies[i]))

}

plot(x, Reduce("+", y_reconstructed), type = "l")
# lines(x, y, col = "red", type = "l", lty = 2)
```

```{r}

```

```{r}
x <- 1:100
y <- sin(2*pi*x)
dftmtx(10)[3, 2]
dftmtx(10)[2, 3]
(as.matrix(dftmtx(10)[3, ])) %*% t(as.matrix(Conj(dftmtx(10)[, 3])))
dim( t(as.matrix(Conj(dftmtx(10)[3, ]))))
dim(as.matrix(dftmtx(10)[3, ]))
```

```{r}

X <- matrix(c(32, 21, 23,
           521,631452, 25251,
           1536, 75, 38), nrow= 3)
f_mat <- dftmtx(3) / sqrt(3)
f_mat_inv <- Conj(f_mat)
X
f_mat %*% f_mat_inv %*% X
```

```{r}
data_slice <- 1:538
n <- 96*2
L <- 96

x <- data_slice
y1 <- cos(2 * pi / 12 * x)  # Первая компонента
y2 <- sin(2 * pi / 48 * x)  # Вторая компонента

# Создаем общий временной ряд
y <- IP_values[data_slice]
y_extended <- y |> extend(L)
x <- 1:length(y_extended)
y <- IP_values[data_slice]

# Выполняем быстрое преобразование Фурье
fft_y <- fft(y)

# Получаем амплитуды и фазы
amplitudes <- Mod(fft_y)
phases <- Arg(fft_y)

# Индексы для частот
n <- length(y)
frequencies <- (0:(n-1)) / n

# Функция для восстановления компонент
reconstruct_fft <- function(frequencies, amplitudes, phases) {
  reconstructed <- list()
  n <- length(amplitudes)
  for (i in 1:(length(amplitudes))) {
    reconstructed[[i]] <-
      amplitudes[i] * cos(2 * pi * frequencies[i] * x + phases[i]) / n * 2
  }
  return(reconstructed)
}

y_reconstructed <- reconstruct_fft(frequencies, amplitudes, phases)

# Строим графики
for (i in 1:(n)){
  plot(x, y_reconstructed[[i]], main = paste(frequencies[i]))
  
}
```

```{r}
set.seed(100)
x <- sin(2*pi/5 * 1:20) + cos(2*pi/2 * 1:20)
L <- 10
X <- hankel(x, L)
X
Ft <- dftmtx(L) / sqrt(L)
Ft_inv <- t(Conj(Ft))
# Ft%*%X[,4:10]
# dftmtx(20) %*% t(Conj(dftmtx(20))) %*% as.matrix(x, nrow = 1) / length(x)
# 
# t(Conj(dftmtx(20))) %*% as.matrix(x, nrow = 1)



# Ft %*%  (Ft_inv %*% X |> diag_averaging())
Ft %*% (Ft_inv %*% X[,5:11] |> apply(1, mean) |> unlist())

dsafads <- function(index_beg){
  Ft %*% (Ft_inv %*% X[, index_beg:(index_beg + dim(X)[1] - 1)] |> unlist())
}

lapply(1:(dim(X)[2] - dim(X)[1]), dsafads)

Ft_inv %*% X 
```

```{r}
Ft_inv_X <- Ft_inv %*% X
f1 <- function(index_beg){
  Ft %*% Ft_inv_X[, index_beg:(index_beg + dim(X)[1] - 1)] |> unlist()
}




```

```{r}
Ft__inv_component_wise <- sweep(Ft_inv, 1, X[, 2], '*')
F_t_compose <- Ft %*% Ft__inv_component_wise
F_t_compose
for (i in 1:L) {
  plot(1:L, F_t_compose[, i])
}

sum(F_t_compose[,3])
x[2]
```

### CiSSA через Фурье

```{r}
# n <- 96*2-1
# x <- 0:(n-1)
# L <- 96
# y <- sin(2*pi/12 * x)
# 
# X <- hankel(y, L)
# 
# Ft <- dftmtx(L) / sqrt(L)
# Ft_inv <- t(Conj(Ft))
# 
# component_wise_mult <- function(index){
#   Ft %*% t(sweep(Ft_inv, 1, X[, index], '*'))
# }
# 
# 
# averaging <- function(res_comp_wise_mult){
#   K <- dim(X)[2]
#   counters <- rep(0, n)
#   res <- matrix(0, nrow = n, ncol = L)
#   for (i in 1:K){
#     res[i:(i+L-1), ] <- res[i:(i+L-1), ] + res_comp_wise_mult[[i]]
#     counters[i:(i+L-1)] <- counters[i:(i+L-1)] + 1
#   }
#   for (i in 1:n){
#     res[i, ] <- res[i, ] / counters[i]
#   }
#   res
# }
# 
# 
# res_comp_wise <- lapply(1:L, component_wise_mult)
# print(res_comp_wise[[1]])
# 
# 
# plot(1:L, res_comp_wise[[1]][, 2])
# 
# avr <- averaging(res_comp_wise)
# 
# for (i in 1:dim(res$t_series)[2]){
#   plot(x, avr[i, ])
# }
# 
# # plot(x, rowSums(avr))
# 
# group_by_elementary_freq_foureir <- function(res_averaged){
#   nf2 <- 0
#   if (L %% 2) {
#     nf2 <- (L + 1) / 2 - 1
#   } else {
#     nf2 <- L / 2 - 1
#   }
#   nft <- nf2 + abs((L %% 2) - 2)
#   
#   Z <- matrix(0, ncol = nft, nrow = n)
#   
#   # print(Z |> dim())
#   # print(res_averaged |> dim())
#   
#   Z[, 1] <- res_averaged[, 1]
#   for (k in 1:nf2) {
#     Z[, k + 1] <- res_averaged[, k + 1] + res_averaged[, L + 2 - (k + 1)]
#   }
#   if (L %% 2 != 0) {
#     Z[, nft] <- res_averaged[, nft]
#   }
#   
#   
#   return(list(
#     t_series = Z,
#     freq = (0:dim(Z)[2])/L
#   ))
# }
# 
# res <- group_by_elementary_freq_foureir(avr)
# 
# # print(res$t_series |> colSums() |> length())
# # print(x |> length())
# 
# # plot(x, rowSums(res$t_series) )
# # plot(x, y)
# 
# t_series <- res$t_series
# 
# plot(x, t_series[, 1])
# 
# for (i in 1:dim(res$t_series)[2]){
#   plot(x, res$t_series[, i])
# }

```

```{r}
# Ft %*% sweep(Ft_inv, 1, X[, 1], '*')[2, ]
```

```{r}
reconstruct_fft <- function(x, y, frequencies) {
  # Выполняем быстрое преобразование Фурье
  fft_y <- fft(y)

  # Получаем амплитуды и фазы
  amplitudes <- Mod(fft_y)
  phases <- Arg(fft_y)
  
  reconstructed <- matrix(0, length(amplitudes), length(x))
  n <- length(amplitudes)
  for (i in 1:(length(amplitudes))) {
    # print(i)
    reconstructed[i, ] <-
      amplitudes[i] * 
      cos(2 * pi * frequencies[i] * (x) + phases[i]) /
      n
  }
  # plot(x, reconstructed[9, ], main = paste(x[1]), type = "l", col = "red")
  # lines(x, y/2)
  # plot(x, reconstructed[L+2-9, ], main = paste(x[1]))
  return(reconstructed)
}


n <- 96*2-1
x <- 0:(n-1)
L <- 48
frequencies <- (0:(L-1)) / L
y <- sin(2*pi/12 * x)
y_main <- y
x_main <- x

X <- hankel(y, L)
K <- dim(X)[2]
res <- list()
for (i in 1:K){
  y <- X[, i]
  x <- x_main[1:(1 + L - 1)]
  res[[i]] <- reconstruct_fft(x, y, frequencies)
}

# for (i in 1:L){
#   plot(x, res[[i]][9, ])
# }

res_mult <- res
# res

averaging <- function(res_comp_wise_mult){
  K <- dim(X)[2]
  counters <- rep(0, n)
  res <- matrix(0, ncol = n, nrow = L)
  for (i in 1:length(res_comp_wise_mult)){
    res[, i:(i+L-1)] <- res[, i:(i+L-1)] + res_comp_wise_mult[[i]]
    counters[i:(i+L-1)] <- counters[i:(i+L-1)] + 1
  }
  for (i in 1:n){
    res[, i] <- res[, i] / counters[i]
  }
  res
}

avr <- averaging(res_mult)

group_by_elementary_freq_foureir <- function(res_averaged){
  nf2 <- 0
  if (L %% 2) {
    nf2 <- (L + 1) / 2 - 1
  } else {
    nf2 <- L / 2 - 1
  }
  nft <- nf2 + abs((L %% 2) - 2)
  
  Z <- matrix(0, ncol = nft, nrow = n)
  
  # print(Z |> dim())
  # print(res_averaged |> dim())
  
  Z[, 1] <- res_averaged[1, ]
  for (k in 1:nf2) {
    Z[, k + 1] <- res_averaged[k + 1, ] + res_averaged[L + 2 - (k + 1), ]
  }
  if (L %% 2 != 0) {
    Z[, nft] <- res_averaged[nft, ]
  }
  
  
  return(list(
    t_series = Z,
    freq = (0:dim(Z)[2])/L
  ))
}


rs <- group_by_elementary_freq_foureir(avr)

plot(x_main, rs$t_series[, 5])

```

#### IP_values

```{r}
data_slice <- 1:537
dates_slice <- dates[data_slice]
IP_values_slice <- IP_values[data_slice]
eps <- 1/193


c <- circulant_SSA(IP_values_slice, L = 192, extend_flag = FALSE)
r <- c$t_series






reconstruct_fft <- function(x, y, frequencies) {
  # Выполняем быстрое преобразование Фурье
  fft_y <- fft(y)

  # Получаем амплитуды и фазы
  amplitudes <- Mod(fft_y)
  phases <- Arg(fft_y)
  
  reconstructed <- matrix(0, length(amplitudes), length(x))
  n <- length(amplitudes)
  for (i in 1:(length(amplitudes))) {
    # print(i)
    reconstructed[i, ] <-
      amplitudes[i] * 
      cos(2 * pi * frequencies[i] * (x) + phases[i]) /
      n
  }
  # plot(x, reconstructed[9, ], main = paste(x[1]), type = "l", col = "red")
  # lines(x, y/2)
  # plot(x, reconstructed[L+2-9, ], main = paste(x[1]))
  return(reconstructed)
}


n <- 537
x <- 0:(n-1)
L <- 192
frequencies <- (0:(L-1)) / L
y <- IP_values_slice
y_main <- y
x_main <- x

X <- hankel(y, L)
K <- dim(X)[2]
res <- list()
for (i in 1:K){
  y <- X[, i]
  x <- x_main[1:(1 + L - 1)]
  res[[i]] <- reconstruct_fft(x, y, frequencies)
}

# for (i in 1:L){
#   plot(x, res[[i]][9, ])
# }

res_mult <- res
# res

averaging <- function(res_comp_wise_mult){
  K <- dim(X)[2]
  counters <- rep(0, n)
  res <- matrix(0, ncol = n, nrow = L)
  for (i in 1:length(res_comp_wise_mult)){
    res[, i:(i+L-1)] <- res[, i:(i+L-1)] + res_comp_wise_mult[[i]]
    counters[i:(i+L-1)] <- counters[i:(i+L-1)] + 1
  }
  for (i in 1:n){
    res[, i] <- res[, i] / counters[i]
  }
  res
}

avr <- averaging(res_mult)

group_by_elementary_freq_foureir <- function(res_averaged){
  nf2 <- 0
  if (L %% 2) {
    nf2 <- (L + 1) / 2 - 1
  } else {
    nf2 <- L / 2 - 1
  }
  nft <- nf2 + abs((L %% 2) - 2)
  
  Z <- matrix(0, ncol = nft, nrow = n)
  
  # print(Z |> dim())
  # print(res_averaged |> dim())
  
  Z[, 1] <- res_averaged[1, ]
  for (k in 1:nf2) {
    Z[, k + 1] <- res_averaged[k + 1, ] + res_averaged[L + 2 - (k + 1), ]
  }
  if (L %% 2 != 0) {
    Z[, nft] <- res_averaged[nft, ]
  }
  
  
  return(list(
    t_series = Z,
    freq = (0:dim(Z)[2])/L
  ))
}


rs <- group_by_elementary_freq_foureir(avr)

# plot(x_main, rs$t_series[, 5])


for (i in 1:dim(r)[2]){
  plot(1:n, rs$t_series[, i], col= "red", type = "l")
  lines(1:n, r[, i])
}


```

```{r}
cissa_like_fourier_transform <- function(ts, L){
  reconstruct_fft <- function(x, y, frequencies) {
    # Выполняем быстрое преобразование Фурье
    fft_y <- fft(y)
  
    # Получаем амплитуды и фазы
    amplitudes <- Mod(fft_y)
    phases <- Arg(fft_y)
    
    reconstructed <- matrix(0, length(amplitudes), length(x))
    n <- length(amplitudes)
    for (i in 1:(length(amplitudes))) {
      # print(i)
      reconstructed[i, ] <-
        amplitudes[i] * 
        cos(2 * pi * frequencies[i] * (x) + phases[i]) /
        n
    }
    # plot(x, reconstructed[9, ], main = paste(x[1]), type = "l", col = "red")
    # lines(x, y/2)
    # plot(x, reconstructed[L+2-9, ], main = paste(x[1]))
    return(reconstructed)
  }
  
  
  n <- length(ts)
  x <- 0:(n-1)
  L <- L
  frequencies <- (0:(L-1)) / L
  y <- ts
  y_main <- y
  x_main <- x
  
  X <- hankel(y, L)
  K <- dim(X)[2]
  res <- list()
  for (i in 1:K){
    y <- X[, i]
    x <- x_main[1:(1 + L - 1)]
    res[[i]] <- reconstruct_fft(x, y, frequencies)
  }
  
  # for (i in 1:L){
  #   plot(x, res[[i]][9, ])
  # }
  
  res_mult <- res
  # res
  
  averaging <- function(res_comp_wise_mult){
    K <- dim(X)[2]
    counters <- rep(0, n)
    res <- matrix(0, ncol = n, nrow = L)
    for (i in 1:length(res_comp_wise_mult)){
      res[, i:(i+L-1)] <- res[, i:(i+L-1)] + res_comp_wise_mult[[i]]
      counters[i:(i+L-1)] <- counters[i:(i+L-1)] + 1
    }
    for (i in 1:n){
      res[, i] <- res[, i] / counters[i]
    }
    res
  }
  
  avr <- averaging(res_mult)
  
  group_by_elementary_freq_foureir <- function(res_averaged){
    nf2 <- 0
    if (L %% 2) {
      nf2 <- (L + 1) / 2 - 1
    } else {
      nf2 <- L / 2 - 1
    }
    nft <- nf2 + abs((L %% 2) - 2)
    
    Z <- matrix(0, ncol = nft, nrow = n)
    
    # print(Z |> dim())
    # print(res_averaged |> dim())
    
    Z[, 1] <- res_averaged[1, ]
    for (k in 1:nf2) {
      Z[, k + 1] <- res_averaged[k + 1, ] + res_averaged[L + 2 - (k + 1), ]
    }
    if (L %% 2 != 0) {
      Z[, nft] <- res_averaged[nft, ]
    }
    
    
    return(list(
      t_series = Z,
      freq = (0:dim(Z)[2])/L
    ))
  }


  rs <- group_by_elementary_freq_foureir(avr)
  return(rs)
}

```

```{r}
data_slice <- 1:537
dates_slice <- dates[data_slice]
IP_values_slice <- IP_values[data_slice]
eps <- 1/193


c <- circulant_SSA(IP_values_slice, L = 192, extend_flag = FALSE)
r <- c$t_series

c_ft <- cissa_like_fourier_transform(IP_values_slice, L = 192)
r_ft <- c_ft$t_series


for (i in 1:dim(r)[2]){
  plot(1:n, r_ft[, i], col= "red", type = "l", lwd = 2)
  lines(1:n, r[, i])
}
```
